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Properties

Label 2-725-145.17-c1-0-22
Degree $2$
Conductor $725$
Sign $0.934 + 0.355i$
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  − 2.23·2-s + 3.00·4-s + (2.23 + 2.23i)7-s − 2.23·8-s + 3·9-s + (4 − 4i)11-s + (−4.47 − 4.47i)13-s + (−5.00 − 5.00i)14-s − 0.999·16-s + 4.47·17-s − 6.70·18-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  − 1.58·2-s + 1.50·4-s + (0.845 + 0.845i)7-s − 0.790·8-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 + 1.08·17-s − 1.58·18-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.934 + 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849547 - 0.156049i\)
\(L(\frac12)\) \(\approx\) \(0.849547 - 0.156049i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 2T^{2} \)
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 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 + (2.23 - 2.23i)T - 23iT^{2} \)
31 \( 1 + (4 - 4i)T - 31iT^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 4.47iT - 43T^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
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 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 + (-2 - 2i)T + 79iT^{2} \)
83 \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \)
89 \( 1 + (-7 - 7i)T + 89iT^{2} \)
97 \( 1 - 4.47iT - 97T^{2} \)
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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

−10.11314811405201076562339695722, −9.477162418801577634192277542645, −8.698419388197151665336653787005, −7.81313517625536148431415805889, −7.41398140310428033778262602195, −6.06227702888113593816081160643, −5.14550684794159414929233760858, −3.57290454014270065124473920355, −2.06555761095676407786282457783, −0.942753274648310271344165429892, 1.24785424071002230282571312133, 1.98770239337389398298616446257, 4.12150757926765518937995571478, 4.78580527348481998391933332409, 6.71385539111449231587168571173, 7.29900167194767675144871450527, 7.66418491595981960244626955183, 8.967326600151403505434077369813, 9.697531610731478465954669893318, 10.03095069446887283500695752542

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