Properties

Label 4-1815e2-1.1-c1e2-0-18
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 4-s − 2·5-s − 4·6-s + 4·7-s + 3·9-s − 4·10-s − 2·12-s + 8·14-s + 4·15-s + 16-s + 8·17-s + 6·18-s + 8·19-s − 2·20-s − 8·21-s − 8·23-s + 3·25-s − 4·27-s + 4·28-s + 4·29-s + 8·30-s − 2·32-s + 16·34-s − 8·35-s + 3·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s + 9-s − 1.26·10-s − 0.577·12-s + 2.13·14-s + 1.03·15-s + 1/4·16-s + 1.94·17-s + 1.41·18-s + 1.83·19-s − 0.447·20-s − 1.74·21-s − 1.66·23-s + 3/5·25-s − 0.769·27-s + 0.755·28-s + 0.742·29-s + 1.46·30-s − 0.353·32-s + 2.74·34-s − 1.35·35-s + 1/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.969843165\)
\(L(\frac12)\) \(\approx\) \(3.969843165\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.412082325762997804166663668694, −9.391205086302685285425505423988, −8.211634952413091386462743891474, −8.128488586996754237503935423550, −7.928497381202798979266300606782, −7.37413424425016596565919742829, −7.25634309139352639933884134104, −6.45866738120684513238330216415, −5.94035409344740250740227525103, −5.76355159345988795424692589823, −5.21415867685166717030687678638, −4.95841773275613071423954266744, −4.68350005169791020883611443321, −4.24952058157792743491051886228, −3.63156436025573235487389819775, −3.57082737147522662215961429392, −2.76623573706322311179080398298, −1.92598098534134565080600944538, −1.17994191158327235575689572230, −0.77040743907051245249807996498, 0.77040743907051245249807996498, 1.17994191158327235575689572230, 1.92598098534134565080600944538, 2.76623573706322311179080398298, 3.57082737147522662215961429392, 3.63156436025573235487389819775, 4.24952058157792743491051886228, 4.68350005169791020883611443321, 4.95841773275613071423954266744, 5.21415867685166717030687678638, 5.76355159345988795424692589823, 5.94035409344740250740227525103, 6.45866738120684513238330216415, 7.25634309139352639933884134104, 7.37413424425016596565919742829, 7.928497381202798979266300606782, 8.128488586996754237503935423550, 8.211634952413091386462743891474, 9.391205086302685285425505423988, 9.412082325762997804166663668694

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