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Properties

Label 2-425-85.59-c1-0-9
Degree $2$
Conductor $425$
Sign $0.905 - 0.424i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (0.187 + 0.187i)2-s + (1.67 + 0.692i)3-s − 1.92i·4-s + (0.183 + 0.443i)6-s + (1.88 + 4.55i)7-s + (0.737 − 0.737i)8-s + (0.193 + 0.193i)9-s + (1.50 + 3.62i)11-s + (1.33 − 3.22i)12-s − 2.07·13-s + (−0.500 + 1.20i)14-s − 3.58·16-s + (3.19 − 2.60i)17-s + 0.0724i·18-s + (2.08 − 2.08i)19-s + ⋯
L(s)  = 1  + (0.132 + 0.132i)2-s + (0.965 + 0.399i)3-s − 0.964i·4-s + (0.0749 + 0.181i)6-s + (0.712 + 1.72i)7-s + (0.260 − 0.260i)8-s + (0.0643 + 0.0643i)9-s + (0.452 + 1.09i)11-s + (0.385 − 0.931i)12-s − 0.574·13-s + (−0.133 + 0.322i)14-s − 0.895·16-s + (0.774 − 0.632i)17-s + 0.0170i·18-s + (0.478 − 0.478i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06063 + 0.459262i\)
\(L(\frac12)\) \(\approx\) \(2.06063 + 0.459262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-3.19 + 2.60i)T \)
good2 \( 1 + (-0.187 - 0.187i)T + 2iT^{2} \)
3 \( 1 + (-1.67 - 0.692i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.88 - 4.55i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.50 - 3.62i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
19 \( 1 + (-2.08 + 2.08i)T - 19iT^{2} \)
23 \( 1 + (-3.58 + 1.48i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (3.22 + 1.33i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.23 + 5.40i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.88 - 0.781i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (10.9 - 4.53i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.91 - 3.91i)T - 43iT^{2} \)
47 \( 1 - 0.453T + 47T^{2} \)
53 \( 1 + (4.60 + 4.60i)T + 53iT^{2} \)
59 \( 1 + (4.92 + 4.92i)T + 59iT^{2} \)
61 \( 1 + (0.0429 - 0.0177i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 + (-5.85 + 14.1i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.36 - 8.12i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.991 - 2.39i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-1.09 - 1.09i)T + 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + (-4.07 + 9.83i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.43994162221925196446847665223, −9.836690255785868884321173755802, −9.532184480758723548532244556726, −8.737869730365920918948175381110, −7.69667715795434720304308103422, −6.43526397990775080776048439553, −5.28721668235906179083098737706, −4.65297434152227968381561248336, −2.90199890300939264194537822132, −1.89914338663940957540077541034, 1.51149568025201142211707020345, 3.22043460509630503315525633359, 3.76076808004203484857378622314, 5.09364683911584494416424922718, 6.88166058986387754114295195860, 7.64062707622858507009144247302, 8.153664336305076514741308141030, 9.005291885841410862586449785574, 10.36362286428967090563915268301, 11.14030006319458540483479804287

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