Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \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  + 2.54·2-s − 2.46·3-s + 4.48·4-s − 6.26·6-s − 7-s + 6.31·8-s + 3.06·9-s − 4.70·11-s − 11.0·12-s + 2.32·13-s − 2.54·14-s + 7.11·16-s + 1.82·17-s + 7.80·18-s + 7.09·19-s + 2.46·21-s − 11.9·22-s + 23-s − 15.5·24-s + 5.92·26-s − 0.159·27-s − 4.48·28-s − 9.98·29-s + 3.53·31-s + 5.48·32-s + 11.5·33-s + 4.64·34-s + ⋯
L(s)  = 1  + 1.80·2-s − 1.42·3-s + 2.24·4-s − 2.55·6-s − 0.377·7-s + 2.23·8-s + 1.02·9-s − 1.41·11-s − 3.18·12-s + 0.645·13-s − 0.680·14-s + 1.77·16-s + 0.443·17-s + 1.83·18-s + 1.62·19-s + 0.537·21-s − 2.55·22-s + 0.208·23-s − 3.17·24-s + 1.16·26-s − 0.0307·27-s − 0.846·28-s − 1.85·29-s + 0.635·31-s + 0.969·32-s + 2.01·33-s + 0.797·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.394347366$
$L(\frac12)$  $\approx$  $3.394347366$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 - 2.54T + 2T^{2} \)
3 \( 1 + 2.46T + 3T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 - 2.32T + 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
29 \( 1 + 9.98T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 - 0.166T + 37T^{2} \)
41 \( 1 - 7.25T + 41T^{2} \)
43 \( 1 - 9.57T + 43T^{2} \)
47 \( 1 + 4.66T + 47T^{2} \)
53 \( 1 - 0.961T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 0.954T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 - 4.59T + 71T^{2} \)
73 \( 1 - 7.59T + 73T^{2} \)
79 \( 1 - 5.73T + 79T^{2} \)
83 \( 1 - 5.57T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−7.979530895472787558413998424473, −7.31570602352334414295290647492, −6.63647011268521973967109058209, −5.71987581058712131890358150037, −5.55147688013980852460921687779, −4.97833642844125336051173883878, −3.99592888680963863602200642469, −3.23888884765524645645014267858, −2.33091576500480927815931154436, −0.861909461123922764920358436227, 0.861909461123922764920358436227, 2.33091576500480927815931154436, 3.23888884765524645645014267858, 3.99592888680963863602200642469, 4.97833642844125336051173883878, 5.55147688013980852460921687779, 5.71987581058712131890358150037, 6.63647011268521973967109058209, 7.31570602352334414295290647492, 7.979530895472787558413998424473

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