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Properties

Label 2-425-85.4-c1-0-7
Degree $2$
Conductor $425$
Sign $0.587 - 0.809i$
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  − 1.12·2-s + (1.75 + 1.75i)3-s − 0.729·4-s + (−1.97 − 1.97i)6-s + (1.72 − 1.72i)7-s + 3.07·8-s + 3.16i·9-s + (2.57 + 2.57i)11-s + (−1.28 − 1.28i)12-s − 3.64i·13-s + (−1.94 + 1.94i)14-s − 2.00·16-s + (2.79 + 3.03i)17-s − 3.56i·18-s + 2.61i·19-s + ⋯
L(s)  = 1  − 0.796·2-s + (1.01 + 1.01i)3-s − 0.364·4-s + (−0.807 − 0.807i)6-s + (0.652 − 0.652i)7-s + 1.08·8-s + 1.05i·9-s + (0.775 + 0.775i)11-s + (−0.369 − 0.369i)12-s − 1.01i·13-s + (−0.520 + 0.520i)14-s − 0.502·16-s + (0.677 + 0.735i)17-s − 0.840i·18-s + 0.599i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.587 - 0.809i$
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: \((2,\ 425,\ (\ :1/2),\ 0.587 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13632 + 0.579471i\)
\(L(\frac12)\) \(\approx\) \(1.13632 + 0.579471i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.79 - 3.03i)T \)
good2 \( 1 + 1.12T + 2T^{2} \)
3 \( 1 + (-1.75 - 1.75i)T + 3iT^{2} \)
7 \( 1 + (-1.72 + 1.72i)T - 7iT^{2} \)
11 \( 1 + (-2.57 - 2.57i)T + 11iT^{2} \)
13 \( 1 + 3.64iT - 13T^{2} \)
19 \( 1 - 2.61iT - 19T^{2} \)
23 \( 1 + (-0.993 + 0.993i)T - 23iT^{2} \)
29 \( 1 + (0.601 - 0.601i)T - 29iT^{2} \)
31 \( 1 + (6.67 - 6.67i)T - 31iT^{2} \)
37 \( 1 + (-7.78 - 7.78i)T + 37iT^{2} \)
41 \( 1 + (6.74 + 6.74i)T + 41iT^{2} \)
43 \( 1 + 7.47T + 43T^{2} \)
47 \( 1 + 5.42iT - 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 1.40iT - 59T^{2} \)
61 \( 1 + (-0.804 - 0.804i)T + 61iT^{2} \)
67 \( 1 + 2.07iT - 67T^{2} \)
71 \( 1 + (-8.69 + 8.69i)T - 71iT^{2} \)
73 \( 1 + (1.04 + 1.04i)T + 73iT^{2} \)
79 \( 1 + (6.34 + 6.34i)T + 79iT^{2} \)
83 \( 1 + 2.52T + 83T^{2} \)
89 \( 1 - 1.66T + 89T^{2} \)
97 \( 1 + (8.67 + 8.67i)T + 97iT^{2} \)
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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

−10.66233092643000456334175405511, −10.23476905696706962649722815907, −9.549152981250602679063676154243, −8.581017875013666177869256372232, −8.093679424519897464697996450570, −7.10846361215202124584124110375, −5.23725311522524925381536554277, −4.25305704806141476845566387557, −3.46295834246923675864236430135, −1.52469581997684026580208538731, 1.21466502385630849746773851360, 2.36863978652721378554944774354, 3.91189440311182328895623022895, 5.33215637939031244556350617931, 6.78250995675399693756444793003, 7.65093500335397419028579149146, 8.392348322412645083146524810534, 9.075924835090840917970591137403, 9.594061649081853944558351596640, 11.19401428807679805022700740138

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