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Properties

Label 2-425-17.13-c1-0-5
Degree $2$
Conductor $425$
Sign $-0.943 + 0.329i$
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  + 2.38i·2-s + (2.23 + 2.23i)3-s − 3.69·4-s + (−5.32 + 5.32i)6-s + (0.155 − 0.155i)7-s − 4.04i·8-s + 6.95i·9-s + (0.371 − 0.371i)11-s + (−8.24 − 8.24i)12-s + 1.96·13-s + (0.371 + 0.371i)14-s + 2.25·16-s + (2.23 − 3.46i)17-s − 16.5·18-s − 4i·19-s + ⋯
L(s)  = 1  + 1.68i·2-s + (1.28 + 1.28i)3-s − 1.84·4-s + (−2.17 + 2.17i)6-s + (0.0587 − 0.0587i)7-s − 1.42i·8-s + 2.31i·9-s + (0.111 − 0.111i)11-s + (−2.37 − 2.37i)12-s + 0.545·13-s + (0.0992 + 0.0992i)14-s + 0.564·16-s + (0.541 − 0.841i)17-s − 3.90·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.943 + 0.329i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.943 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319561 - 1.88280i\)
\(L(\frac12)\) \(\approx\) \(0.319561 - 1.88280i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 3.46i)T \)
good2 \( 1 - 2.38iT - 2T^{2} \)
3 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
7 \( 1 + (-0.155 + 0.155i)T - 7iT^{2} \)
11 \( 1 + (-0.371 + 0.371i)T - 11iT^{2} \)
13 \( 1 - 1.96T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-0.263 + 0.263i)T - 23iT^{2} \)
29 \( 1 + (4.95 + 4.95i)T + 29iT^{2} \)
31 \( 1 + (-2.06 - 2.06i)T + 31iT^{2} \)
37 \( 1 + (-4.04 - 4.04i)T + 37iT^{2} \)
41 \( 1 + (-0.563 + 0.563i)T - 41iT^{2} \)
43 \( 1 - 2.49iT - 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 + 5.92iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (4 - 4i)T - 61iT^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + (5.06 + 5.06i)T + 71iT^{2} \)
73 \( 1 + (0.838 + 0.838i)T + 73iT^{2} \)
79 \( 1 + (-4.75 + 4.75i)T - 79iT^{2} \)
83 \( 1 + 6.11iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (-6.51 - 6.51i)T + 97iT^{2} \)
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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

−11.46399986626339427823147994779, −10.32628816911807241137316580284, −9.332564048725930912024306807610, −8.932918387721317734596934602406, −7.997843342667142541941950478889, −7.33251804425085837030741581900, −5.98067506605952785867493403897, −4.89501370821803523404062536700, −4.15495200050840558693155687476, −2.88262674960254587921120167244, 1.25158081202312994184748756658, 2.10005423651942766073057479027, 3.24910307374561654247165521117, 3.98910942136500237746214171943, 5.93039580082884512041588446735, 7.26851443340741312296706921776, 8.237872596272266127280189759874, 8.933697626750876846239980970620, 9.767246669565860332771412089591, 10.77479824601101696107137665483

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