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Properties

Label 2-425-85.19-c1-0-15
Degree $2$
Conductor $425$
Sign $0.772 + 0.635i$
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 + (−0.0424 − 0.102i)3-s + 0.0682i·4-s + (−0.142 − 0.0589i)6-s + (−1.58 − 0.656i)7-s + (2.03 + 2.03i)8-s + (2.11 − 2.11i)9-s + (5.35 + 2.21i)11-s + (0.00698 − 0.00289i)12-s + 1.25·13-s + (−2.20 + 0.912i)14-s + 3.85·16-s + (−1.85 − 3.68i)17-s − 4.15i·18-s + (−1.99 − 1.99i)19-s + ⋯
L(s)  = 1  + (0.694 − 0.694i)2-s + (−0.0244 − 0.0591i)3-s + 0.0341i·4-s + (−0.0580 − 0.0240i)6-s + (−0.599 − 0.248i)7-s + (0.718 + 0.718i)8-s + (0.704 − 0.704i)9-s + (1.61 + 0.669i)11-s + (0.00201 − 0.000835i)12-s + 0.347·13-s + (−0.588 + 0.243i)14-s + 0.964·16-s + (−0.448 − 0.893i)17-s − 0.978i·18-s + (−0.458 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.772 + 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98927 - 0.713055i\)
\(L(\frac12)\) \(\approx\) \(1.98927 - 0.713055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 3.68i)T \)
good2 \( 1 + (-0.982 + 0.982i)T - 2iT^{2} \)
3 \( 1 + (0.0424 + 0.102i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.58 + 0.656i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-5.35 - 2.21i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.25T + 13T^{2} \)
19 \( 1 + (1.99 + 1.99i)T + 19iT^{2} \)
23 \( 1 + (-0.785 + 1.89i)T + (-16.2 - 16.2i)T^{2} \)
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 \( 1 + (-2.41 - 5.82i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.61 - 8.73i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.25 + 5.25i)T + 43iT^{2} \)
47 \( 1 + 7.63T + 47T^{2} \)
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 \( 1 - 2.46iT - 67T^{2} \)
71 \( 1 + (12.4 - 5.15i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.39 - 2.23i)T + (51.6 - 51.6i)T^{2} \)
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 \( 1 + (7.84 - 3.25i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.49181935550208185912154332380, −10.24927062147128939516141076123, −9.438457798369610890763181017625, −8.503040298703796359056382464654, −6.93672746574944323324541586535, −6.64699800510240495623251416828, −4.84525482252253528026800425371, −4.03091502583069154309334722018, −3.13318857323598899728981471405, −1.52737411149011342561817659473, 1.59763802934284812315497273609, 3.69657041359912919872687014615, 4.40867470562022159623757388853, 5.80792455177091860779132396496, 6.34582838420603282899530445620, 7.25267406844730023818041650471, 8.476223945775516605323751090354, 9.486598586559164166390320731168, 10.36878452308133548545275677310, 11.25717993426851260855089986242

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