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  − 1.23·2-s − 2.68·3-s − 0.466·4-s + 3.32·6-s − 7-s + 3.05·8-s + 4.22·9-s − 0.846·11-s + 1.25·12-s − 2.55·13-s + 1.23·14-s − 2.84·16-s + 7.07·17-s − 5.23·18-s − 0.476·19-s + 2.68·21-s + 1.04·22-s + 23-s − 8.21·24-s + 3.15·26-s − 3.30·27-s + 0.466·28-s + 8.63·29-s + 3.31·31-s − 2.58·32-s + 2.27·33-s − 8.75·34-s + ⋯
L(s)  = 1  − 0.875·2-s − 1.55·3-s − 0.233·4-s + 1.35·6-s − 0.377·7-s + 1.07·8-s + 1.40·9-s − 0.255·11-s + 0.362·12-s − 0.707·13-s + 0.330·14-s − 0.712·16-s + 1.71·17-s − 1.23·18-s − 0.109·19-s + 0.586·21-s + 0.223·22-s + 0.208·23-s − 1.67·24-s + 0.619·26-s − 0.635·27-s + 0.0881·28-s + 1.60·29-s + 0.594·31-s − 0.456·32-s + 0.396·33-s − 1.50·34-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  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4425663585$
$L(\frac12)$  $\approx$  $0.4425663585$
$L(\frac{3}{2})$   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 \( 1 + 1.23T + 2T^{2} \)
3 \( 1 + 2.68T + 3T^{2} \)
11 \( 1 + 0.846T + 11T^{2} \)
13 \( 1 + 2.55T + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 + 0.476T + 19T^{2} \)
29 \( 1 - 8.63T + 29T^{2} \)
31 \( 1 - 3.31T + 31T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + 8.93T + 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 - 9.92T + 71T^{2} \)
73 \( 1 + 7.31T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 3.72T + 83T^{2} \)
89 \( 1 + 8.76T + 89T^{2} \)
97 \( 1 + 1.82T + 97T^{2} \)
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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

−8.410537542440521036930800994057, −7.68889425825621200474655210581, −7.03464078749698564697940091732, −6.26791535185147433723260514332, −5.40843255985375048087511524115, −4.94229437287290300332728074979, −4.09004884747658101621867722359, −2.85076642811841786055464216772, −1.33821148499433837455629140504, −0.53403699993518861149951819962, 0.53403699993518861149951819962, 1.33821148499433837455629140504, 2.85076642811841786055464216772, 4.09004884747658101621867722359, 4.94229437287290300332728074979, 5.40843255985375048087511524115, 6.26791535185147433723260514332, 7.03464078749698564697940091732, 7.68889425825621200474655210581, 8.410537542440521036930800994057

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