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Properties

Label 2-725-145.133-c1-0-1
Degree $2$
Conductor $725$
Sign $0.735 - 0.677i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·2-s − 3.00·4-s + (−2.23 + 2.23i)7-s + 2.23i·8-s − 3·9-s + (4 − 4i)11-s + (−4.47 + 4.47i)13-s + (5.00 + 5.00i)14-s − 0.999·16-s + 4.47i·17-s + 6.70i·18-s + (−2 − 2i)19-s + (−8.94 − 8.94i)22-s + (2.23 + 2.23i)23-s + (10.0 + 10.0i)26-s + ⋯
L(s)  = 1  − 1.58i·2-s − 1.50·4-s + (−0.845 + 0.845i)7-s + 0.790i·8-s − 9-s + (1.20 − 1.20i)11-s + (−1.24 + 1.24i)13-s + (1.33 + 1.33i)14-s − 0.249·16-s + 1.08i·17-s + 1.58i·18-s + (−0.458 − 0.458i)19-s + (−1.90 − 1.90i)22-s + (0.466 + 0.466i)23-s + (1.96 + 1.96i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325142 + 0.126827i\)
\(L(\frac12)\) \(\approx\) \(0.325142 + 0.126827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (-2 - 5i)T \)
good2 \( 1 + 2.23iT - 2T^{2} \)
3 \( 1 + 3T^{2} \)
7 \( 1 + (2.23 - 2.23i)T - 7iT^{2} \)
11 \( 1 + (-4 + 4i)T - 11iT^{2} \)
13 \( 1 + (4.47 - 4.47i)T - 13iT^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 + (2 + 2i)T + 19iT^{2} \)
23 \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \)
31 \( 1 + (4 - 4i)T - 31iT^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 + (8.94 + 8.94i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (9 - 9i)T - 61iT^{2} \)
67 \( 1 + (-2.23 - 2.23i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 4.47iT - 73T^{2} \)
79 \( 1 + (2 + 2i)T + 79iT^{2} \)
83 \( 1 + (6.70 + 6.70i)T + 83iT^{2} \)
89 \( 1 + (7 + 7i)T + 89iT^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74787496649665684930596115633, −9.575607560616617672646146461736, −9.081610485141186498899796640884, −8.556437422848716090148298105365, −6.77370353448409847282737049181, −6.03635344056750740421106512846, −4.75549265831160926968764724472, −3.52895563092999420150973342314, −2.88885835173010484546137915022, −1.69515584243645869724782583054, 0.17321245809921928363541193332, 2.73983846242147603543495171411, 4.18362870055294490420144220124, 5.06820405621799394898790781356, 6.08766534998906983357659954491, 6.85405300086161632701155170251, 7.47765071589904209981988582375, 8.263756627496740894269724211417, 9.521009271522100879532494322157, 9.730467388430761234567699092777

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