Properties

Degree 2
Conductor $ 5 \cdot 23 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5-s + 7-s − 3·9-s − 2·10-s + 2·11-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s − 6·18-s − 2·19-s − 2·20-s + 4·22-s + 23-s + 25-s − 4·26-s + 2·28-s + 7·29-s − 5·31-s − 8·32-s + 6·34-s − 35-s − 6·36-s + 11·37-s − 4·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s − 9-s − 0.632·10-s + 0.603·11-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s − 1.41·18-s − 0.458·19-s − 0.447·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s + 1.29·29-s − 0.898·31-s − 1.41·32-s + 1.02·34-s − 0.169·35-s − 36-s + 1.80·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(115\)    =    \(5 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{115} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 115,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.808038880$
$L(\frac12)$  $\approx$  $1.808038880$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.64520758283061, −18.33306491163889, −17.19977451994537, −16.29549194714674, −14.97340229530014, −14.58063421507655, −13.72067584909168, −12.49614549362548, −11.83553811939399, −10.95256344710710, −9.254856050792654, −7.991219017033405, −6.518294496660438, −5.372739821848146, −4.224843825189705, −2.873119415257358, 2.873119415257358, 4.224843825189705, 5.372739821848146, 6.518294496660438, 7.991219017033405, 9.254856050792654, 10.95256344710710, 11.83553811939399, 12.49614549362548, 13.72067584909168, 14.58063421507655, 14.97340229530014, 16.29549194714674, 17.19977451994537, 18.33306491163889, 19.64520758283061

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