Properties

Label 4-910e2-1.1-c1e2-0-53
Degree $4$
Conductor $828100$
Sign $1$
Analytic cond. $52.8003$
Root an. cond. $2.69562$
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 + 3·4-s + 4·5-s − 4·6-s − 4·8-s + 2·9-s − 8·10-s + 4·11-s + 6·12-s − 6·13-s + 8·15-s + 5·16-s + 8·17-s − 4·18-s + 6·19-s + 12·20-s − 8·22-s − 2·23-s − 8·24-s + 11·25-s + 12·26-s + 6·27-s − 16·30-s + 8·31-s − 6·32-s + 8·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s − 1.63·6-s − 1.41·8-s + 2/3·9-s − 2.52·10-s + 1.20·11-s + 1.73·12-s − 1.66·13-s + 2.06·15-s + 5/4·16-s + 1.94·17-s − 0.942·18-s + 1.37·19-s + 2.68·20-s − 1.70·22-s − 0.417·23-s − 1.63·24-s + 11/5·25-s + 2.35·26-s + 1.15·27-s − 2.92·30-s + 1.43·31-s − 1.06·32-s + 1.39·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(828100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(52.8003\)
Root analytic conductor: \(2.69562\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 828100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.896107029\)
\(L(\frac12)\) \(\approx\) \(2.896107029\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.909019313289954528911472144695, −9.863799273715597252472957019026, −9.438155601454903147557820918142, −9.258725652792588385750377375437, −8.785464947559925019730791366086, −8.290260678100921602776673099097, −7.896124883047979078406510087824, −7.44142111514652985071167618170, −7.08275830103697089805721086852, −6.65620813428663314754936913889, −5.92669686253139785541748268046, −5.91966122199761279484072653741, −4.91436355291107972404741626140, −4.84733197437917578494953704639, −3.57838092787783101088530829958, −3.21436558443299442175368177187, −2.53101598790025368461818334848, −2.34352637770315376767693448059, −1.27832549824080320124244507217, −1.21245014815265443091380975368, 1.21245014815265443091380975368, 1.27832549824080320124244507217, 2.34352637770315376767693448059, 2.53101598790025368461818334848, 3.21436558443299442175368177187, 3.57838092787783101088530829958, 4.84733197437917578494953704639, 4.91436355291107972404741626140, 5.91966122199761279484072653741, 5.92669686253139785541748268046, 6.65620813428663314754936913889, 7.08275830103697089805721086852, 7.44142111514652985071167618170, 7.896124883047979078406510087824, 8.290260678100921602776673099097, 8.785464947559925019730791366086, 9.258725652792588385750377375437, 9.438155601454903147557820918142, 9.863799273715597252472957019026, 9.909019313289954528911472144695

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