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Properties

Label 2-425-85.4-c1-0-1
Degree $2$
Conductor $425$
Sign $-0.715 - 0.698i$
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  − 1.68·2-s + (1.27 + 1.27i)3-s + 0.830·4-s + (−2.14 − 2.14i)6-s + (−1.92 + 1.92i)7-s + 1.96·8-s + 0.249i·9-s + (−0.0173 − 0.0173i)11-s + (1.05 + 1.05i)12-s + 3.62i·13-s + (3.24 − 3.24i)14-s − 4.97·16-s + (−1.33 + 3.90i)17-s − 0.419i·18-s + 0.603i·19-s + ⋯
L(s)  = 1  − 1.18·2-s + (0.735 + 0.735i)3-s + 0.415·4-s + (−0.875 − 0.875i)6-s + (−0.728 + 0.728i)7-s + 0.695·8-s + 0.0830i·9-s + (−0.00524 − 0.00524i)11-s + (0.305 + 0.305i)12-s + 1.00i·13-s + (0.866 − 0.866i)14-s − 1.24·16-s + (−0.323 + 0.946i)17-s − 0.0987i·18-s + 0.138i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.715 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233471 + 0.573405i\)
\(L(\frac12)\) \(\approx\) \(0.233471 + 0.573405i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 3.90i)T \)
good2 \( 1 + 1.68T + 2T^{2} \)
3 \( 1 + (-1.27 - 1.27i)T + 3iT^{2} \)
7 \( 1 + (1.92 - 1.92i)T - 7iT^{2} \)
11 \( 1 + (0.0173 + 0.0173i)T + 11iT^{2} \)
13 \( 1 - 3.62iT - 13T^{2} \)
19 \( 1 - 0.603iT - 19T^{2} \)
23 \( 1 + (1.94 - 1.94i)T - 23iT^{2} \)
29 \( 1 + (3.01 - 3.01i)T - 29iT^{2} \)
31 \( 1 + (0.422 - 0.422i)T - 31iT^{2} \)
37 \( 1 + (7.50 + 7.50i)T + 37iT^{2} \)
41 \( 1 + (-5.07 - 5.07i)T + 41iT^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 2.05T + 53T^{2} \)
59 \( 1 + 0.926iT - 59T^{2} \)
61 \( 1 + (-8.41 - 8.41i)T + 61iT^{2} \)
67 \( 1 + 5.79iT - 67T^{2} \)
71 \( 1 + (3.84 - 3.84i)T - 71iT^{2} \)
73 \( 1 + (-2.88 - 2.88i)T + 73iT^{2} \)
79 \( 1 + (9.68 + 9.68i)T + 79iT^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 7.11T + 89T^{2} \)
97 \( 1 + (5.01 + 5.01i)T + 97iT^{2} \)
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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

−11.15956476697695540022633386476, −10.19537635428840320173251963409, −9.494522155686347926731682150226, −8.965466542297976072982805391049, −8.349501080403954467205771602382, −7.13615552717680897001821013228, −6.04237622672347235279336625531, −4.48098740347292942916305487124, −3.43625946485627262445773308856, −1.93712779623282061614410557531, 0.52843140716334150758935829914, 2.12058401482233122779805431959, 3.47204119304761635433977328086, 5.03390830400647121201246554304, 6.73819892357824997323041796519, 7.35138843831418212795018926836, 8.179649407880813526829423000880, 8.849273127314369142942803102058, 9.951319354447672089714541935819, 10.38353123398178002369810481202

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