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Properties

Label 2-425-17.9-c1-0-12
Degree $2$
Conductor $425$
Sign $0.913 + 0.406i$
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  + (0.982 − 0.982i)2-s + (−0.102 − 0.0424i)3-s + 0.0682i·4-s + (−0.142 + 0.0589i)6-s + (0.656 + 1.58i)7-s + (2.03 + 2.03i)8-s + (−2.11 − 2.11i)9-s + (5.35 − 2.21i)11-s + (0.00289 − 0.00698i)12-s + 1.25i·13-s + (2.20 + 0.912i)14-s + 3.85·16-s + (3.68 + 1.85i)17-s − 4.15·18-s + (1.99 − 1.99i)19-s + ⋯
L(s)  = 1  + (0.694 − 0.694i)2-s + (−0.0591 − 0.0244i)3-s + 0.0341i·4-s + (−0.0580 + 0.0240i)6-s + (0.248 + 0.599i)7-s + (0.718 + 0.718i)8-s + (−0.704 − 0.704i)9-s + (1.61 − 0.669i)11-s + (0.000835 − 0.00201i)12-s + 0.347i·13-s + (0.588 + 0.243i)14-s + 0.964·16-s + (0.893 + 0.448i)17-s − 0.978·18-s + (0.458 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.913 + 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06126 - 0.438032i\)
\(L(\frac12)\) \(\approx\) \(2.06126 - 0.438032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.68 - 1.85i)T \)
good2 \( 1 + (-0.982 + 0.982i)T - 2iT^{2} \)
3 \( 1 + (0.102 + 0.0424i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.656 - 1.58i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-5.35 + 2.21i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.25iT - 13T^{2} \)
19 \( 1 + (-1.99 + 1.99i)T - 19iT^{2} \)
23 \( 1 + (1.89 - 0.785i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.99 - 4.80i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.64 + 1.09i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (5.82 + 2.41i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.61 + 8.73i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.25 + 5.25i)T + 43iT^{2} \)
47 \( 1 - 7.63iT - 47T^{2} \)
53 \( 1 + (5.09 - 5.09i)T - 53iT^{2} \)
59 \( 1 + (1.54 + 1.54i)T + 59iT^{2} \)
61 \( 1 + (-2.19 - 5.30i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 2.46T + 67T^{2} \)
71 \( 1 + (12.4 + 5.15i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.23 + 5.39i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.62 - 1.50i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-6.29 + 6.29i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (3.25 - 7.84i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.57408326804240952239653593835, −10.54760876179308192454892279153, −9.090566115220771815689893975230, −8.709103591773664242067351928034, −7.36900055443559678251220418347, −6.13811972544696826548412513883, −5.26931089124875677267594557513, −3.85423199802912131244405605488, −3.22538781940504893322124686988, −1.64813817300117692692136875802, 1.48060230022180798708184310949, 3.54136763117212357938297153639, 4.59293462324323925370854382630, 5.47896994859908116063979625231, 6.45737623853928105462610579380, 7.34145155743133988672458556009, 8.203888587807946623158810719141, 9.610110432044029100312621994051, 10.23065013605488505903924246786, 11.37996282112270668370778229232

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