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Properties

Label 2-925-37.4-c1-0-19
Degree $2$
Conductor $925$
Sign $0.616 + 0.787i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (−2.22 − 0.391i)2-s + (−0.457 − 2.59i)3-s + (2.89 + 1.05i)4-s + 5.94i·6-s + (0.761 − 0.639i)7-s + (−2.11 − 1.22i)8-s + (−3.71 + 1.35i)9-s + (−2.23 + 3.87i)11-s + (1.41 − 8.00i)12-s + (0.156 − 0.430i)13-s + (−1.94 + 1.12i)14-s + (−0.502 − 0.421i)16-s + (1.31 + 3.61i)17-s + (8.77 − 1.54i)18-s + (6.33 − 1.11i)19-s + ⋯
L(s)  = 1  + (−1.57 − 0.276i)2-s + (−0.264 − 1.49i)3-s + (1.44 + 0.527i)4-s + 2.42i·6-s + (0.287 − 0.241i)7-s + (−0.748 − 0.432i)8-s + (−1.23 + 0.450i)9-s + (−0.674 + 1.16i)11-s + (0.407 − 2.31i)12-s + (0.0434 − 0.119i)13-s + (−0.518 + 0.299i)14-s + (−0.125 − 0.105i)16-s + (0.319 + 0.877i)17-s + (2.06 − 0.364i)18-s + (1.45 − 0.256i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.616 + 0.787i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555153 - 0.270536i\)
\(L(\frac12)\) \(\approx\) \(0.555153 - 0.270536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (5.40 + 2.79i)T \)
good2 \( 1 + (2.22 + 0.391i)T + (1.87 + 0.684i)T^{2} \)
3 \( 1 + (0.457 + 2.59i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.761 + 0.639i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (2.23 - 3.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.156 + 0.430i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.31 - 3.61i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (-6.33 + 1.11i)T + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (1.59 - 0.923i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.71 - 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.28iT - 31T^{2} \)
41 \( 1 + (-10.2 - 3.71i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 4.21iT - 43T^{2} \)
47 \( 1 + (-6.11 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.18 + 1.83i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.99 + 5.95i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.980 - 2.69i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.35 - 1.13i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.98 + 11.2i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + (9.83 + 11.7i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.25 + 0.819i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-0.0452 + 0.0538i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-11.3 + 6.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.936487445542060440530722379397, −9.078754651623100708978455617533, −7.975129833104885796241411105843, −7.66146636844149353319988074179, −7.04498214066533825107367695549, −6.05198477972998714817054221769, −4.79577892603480760113452884253, −2.84247708849283077959951711568, −1.77229384241070351066271506336, −0.975105379583146716307733074544, 0.68592822219262255012927451875, 2.65364000465126123192377851025, 3.86360657042274080824858626428, 5.15223181199993055824886361107, 5.77447022526009096151818759714, 7.06587776398399866106868656810, 8.042001517257204329803436148564, 8.687182955599246163405949888784, 9.427692162429165956840659909433, 10.08778539676968773435213481904

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