Properties

Degree 4
Conductor $ 3^{2} \cdot 13^{2} \cdot 103^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 4·5-s + 4·6-s + 3·9-s + 8·10-s − 2·12-s + 2·13-s + 8·15-s + 16-s + 8·17-s − 6·18-s − 4·20-s + 4·23-s + 4·25-s − 4·26-s − 4·27-s − 4·29-s − 16·30-s − 12·31-s + 2·32-s − 16·34-s + 3·36-s + 8·37-s − 4·39-s + 8·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s + 9-s + 2.52·10-s − 0.577·12-s + 0.554·13-s + 2.06·15-s + 1/4·16-s + 1.94·17-s − 1.41·18-s − 0.894·20-s + 0.834·23-s + 4/5·25-s − 0.784·26-s − 0.769·27-s − 0.742·29-s − 2.92·30-s − 2.15·31-s + 0.353·32-s − 2.74·34-s + 1/2·36-s + 1.31·37-s − 0.640·39-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16136289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16136289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16136289\)    =    \(3^{2} \cdot 13^{2} \cdot 103^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4017} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 16136289,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
103$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 12 T + 168 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.038081041354474494667567838678, −7.87073897975558544031398277480, −7.63801172658218308492118542963, −7.37097537548947672123417905738, −7.16708748858738772212216473851, −6.34065749901947973446936871391, −6.08109876301496222359277774549, −5.85570078586470964474943916454, −5.16156155752868525817998453762, −5.10088895757236015877013821373, −4.27849445892947347507671758975, −4.17677202957386955320901384042, −3.54047749194034220661167476773, −3.39006966306001376982024128932, −2.78290586070126621428328065555, −1.89850389442155214634578328011, −1.06346747357381321427304379993, −1.06300995681247488470159513719, 0, 0, 1.06300995681247488470159513719, 1.06346747357381321427304379993, 1.89850389442155214634578328011, 2.78290586070126621428328065555, 3.39006966306001376982024128932, 3.54047749194034220661167476773, 4.17677202957386955320901384042, 4.27849445892947347507671758975, 5.10088895757236015877013821373, 5.16156155752868525817998453762, 5.85570078586470964474943916454, 6.08109876301496222359277774549, 6.34065749901947973446936871391, 7.16708748858738772212216473851, 7.37097537548947672123417905738, 7.63801172658218308492118542963, 7.87073897975558544031398277480, 8.038081041354474494667567838678

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