Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 31·2-s − 162·3-s + 239·4-s + 1.25e3·5-s − 5.02e3·6-s + 1.41e4·7-s + 899·8-s + 1.96e4·9-s + 3.87e4·10-s − 2.15e4·11-s − 3.87e4·12-s + 2.42e4·13-s + 4.37e5·14-s − 2.02e5·15-s + 8.51e4·16-s − 1.56e5·17-s + 6.10e5·18-s − 9.58e4·19-s + 2.98e5·20-s − 2.28e6·21-s − 6.66e5·22-s − 7.35e5·23-s − 1.45e5·24-s + 1.17e6·25-s + 7.52e5·26-s − 2.12e6·27-s + 3.37e6·28-s + ⋯
L(s)  = 1  + 1.37·2-s − 1.15·3-s + 0.466·4-s + 0.894·5-s − 1.58·6-s + 2.22·7-s + 0.0775·8-s + 9-s + 1.22·10-s − 0.443·11-s − 0.539·12-s + 0.235·13-s + 3.04·14-s − 1.03·15-s + 0.325·16-s − 0.455·17-s + 1.37·18-s − 0.168·19-s + 0.417·20-s − 2.56·21-s − 0.606·22-s − 0.547·23-s − 0.0896·24-s + 3/5·25-s + 0.323·26-s − 0.769·27-s + 1.03·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(225\)    =    \(3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{15} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 225,\ (\ :9/2, 9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(4.43496\)
\(L(\frac12)\)  \(\approx\)  \(4.43496\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + p^{4} T )^{2} \)
5$C_1$ \( ( 1 - p^{4} T )^{2} \)
good2$D_{4}$ \( 1 - 31 T + 361 p T^{2} - 31 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 - 288 p^{2} T + 2107886 p^{2} T^{2} - 288 p^{11} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 + 21512 T + 1790961254 T^{2} + 21512 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 1868 p T + 5870669214 T^{2} - 1868 p^{10} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 + 156956 T + 212670095078 T^{2} + 156956 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 + 95896 T + 629883192438 T^{2} + 95896 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 + 31968 p T + 3363187908526 T^{2} + 31968 p^{10} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 2678212 T + 15397908029438 T^{2} + 2678212 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 - 10782432 T + 69294691361342 T^{2} - 10782432 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 - 21968332 T + 359210373327534 T^{2} - 21968332 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 26060372 T + 693239183881142 T^{2} - 26060372 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 + 7191160 T + 750634586008230 T^{2} + 7191160 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 + 671920 p T + 382042606129310 T^{2} + 671920 p^{10} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 - 3131116 T + 6584849973489806 T^{2} - 3131116 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 + 35494664 T + 7388006896329158 T^{2} + 35494664 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 341497340 T + 52053805546777278 T^{2} - 341497340 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 + 288195816 T + 74605839041196758 T^{2} + 288195816 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 - 210286064 T + 91549406631588686 T^{2} - 210286064 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 232663084 T + 130536207391012086 T^{2} + 232663084 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 + 24755040 T + 43090694479668638 T^{2} + 24755040 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 + 372082152 T + 379908828789982198 T^{2} + 372082152 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 + 427639116 T + 702028302670151638 T^{2} + 427639116 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 1771658884 T + 2207048700436243398 T^{2} - 1771658884 p^{9} T^{3} + p^{18} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.50879103791912782747065835903, −16.99940123368544952783131106395, −16.14276913180374231901880793526, −15.27174069726397164897883445595, −14.39850785303878044264511277679, −14.25474033065510341558678295690, −13.11268265924733439960737401695, −13.09087010916688159790936977715, −11.80263158849163533957508601208, −11.36294620416704591200972006337, −10.69818121115332193298987552172, −9.834405788870403866626550613586, −8.439094873599952929310718602775, −7.57020401659824680483034778008, −6.20036582165188861355970623800, −5.49769070495138489091464465718, −4.72368706125592097975057746691, −4.35095565650486616548205522556, −2.16767088918043568698981287728, −1.08218705022902846707049609956, 1.08218705022902846707049609956, 2.16767088918043568698981287728, 4.35095565650486616548205522556, 4.72368706125592097975057746691, 5.49769070495138489091464465718, 6.20036582165188861355970623800, 7.57020401659824680483034778008, 8.439094873599952929310718602775, 9.834405788870403866626550613586, 10.69818121115332193298987552172, 11.36294620416704591200972006337, 11.80263158849163533957508601208, 13.09087010916688159790936977715, 13.11268265924733439960737401695, 14.25474033065510341558678295690, 14.39850785303878044264511277679, 15.27174069726397164897883445595, 16.14276913180374231901880793526, 16.99940123368544952783131106395, 17.50879103791912782747065835903

Graph of the $Z$-function along the critical line