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Properties

Label 2-425-17.12-c2-0-50
Degree $2$
Conductor $425$
Sign $0.260 + 0.965i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.116 + 0.280i)2-s + (4.44 − 2.97i)3-s + (2.76 − 2.76i)4-s + (1.34 + 0.901i)6-s + (2.10 − 0.418i)7-s + (2.21 + 0.918i)8-s + (7.49 − 18.0i)9-s + (−3.66 + 5.49i)11-s + (4.07 − 20.4i)12-s + (−9.67 − 9.67i)13-s + (0.361 + 0.541i)14-s − 14.9i·16-s + (0.398 + 16.9i)17-s + 5.94·18-s + (4.71 + 11.3i)19-s + ⋯
L(s)  = 1  + (0.0581 + 0.140i)2-s + (1.48 − 0.990i)3-s + (0.690 − 0.690i)4-s + (0.224 + 0.150i)6-s + (0.300 − 0.0597i)7-s + (0.277 + 0.114i)8-s + (0.832 − 2.01i)9-s + (−0.333 + 0.499i)11-s + (0.339 − 1.70i)12-s + (−0.744 − 0.744i)13-s + (0.0258 + 0.0386i)14-s − 0.931i·16-s + (0.0234 + 0.999i)17-s + 0.330·18-s + (0.248 + 0.598i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.68664 - 2.05696i\)
\(L(\frac12)\) \(\approx\) \(2.68664 - 2.05696i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-0.398 - 16.9i)T \)
good2 \( 1 + (-0.116 - 0.280i)T + (-2.82 + 2.82i)T^{2} \)
3 \( 1 + (-4.44 + 2.97i)T + (3.44 - 8.31i)T^{2} \)
7 \( 1 + (-2.10 + 0.418i)T + (45.2 - 18.7i)T^{2} \)
11 \( 1 + (3.66 - 5.49i)T + (-46.3 - 111. i)T^{2} \)
13 \( 1 + (9.67 + 9.67i)T + 169iT^{2} \)
19 \( 1 + (-4.71 - 11.3i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (1.56 + 1.04i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (5.73 - 28.8i)T + (-776. - 321. i)T^{2} \)
31 \( 1 + (-27.7 - 41.5i)T + (-367. + 887. i)T^{2} \)
37 \( 1 + (-39.7 + 26.5i)T + (523. - 1.26e3i)T^{2} \)
41 \( 1 + (74.0 - 14.7i)T + (1.55e3 - 643. i)T^{2} \)
43 \( 1 + (-12.5 + 30.4i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (24.9 + 24.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-10.3 - 25.0i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-69.9 - 28.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-11.8 - 59.6i)T + (-3.43e3 + 1.42e3i)T^{2} \)
67 \( 1 + 88.5iT - 4.48e3T^{2} \)
71 \( 1 + (-26.4 + 17.6i)T + (1.92e3 - 4.65e3i)T^{2} \)
73 \( 1 + (42.8 + 8.52i)T + (4.92e3 + 2.03e3i)T^{2} \)
79 \( 1 + (-23.2 + 34.7i)T + (-2.38e3 - 5.76e3i)T^{2} \)
83 \( 1 + (-87.2 + 36.1i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-62.4 + 62.4i)T - 7.92e3iT^{2} \)
97 \( 1 + (1.86 - 9.38i)T + (-8.69e3 - 3.60e3i)T^{2} \)
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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

−10.52187467768912954927364428444, −9.915518946421370084712627007997, −8.718692392615012726139188080639, −7.86345176594958611037680116365, −7.25257447563716432206533965606, −6.34096468801992767935953812833, −5.04778918306446003494089434967, −3.35666896923421490333847462657, −2.27806073109639831795867067133, −1.34847324862072040566129567342, 2.26391725585698459785470590767, 2.92413273498301134131678261111, 4.04105687690683327266502026369, 4.95280828149080866396557660976, 6.74746640988175330835222599330, 7.84436420175772026024972156642, 8.292485438366261942992000810422, 9.452038486517666610217380548115, 9.967985787526397418860653152267, 11.25749591582068085043886403465

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