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Properties

Label 2-425-17.16-c1-0-17
Degree $2$
Conductor $425$
Sign $0.612 + 0.790i$
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  + 0.311·2-s + 1.12i·3-s − 1.90·4-s + 0.349i·6-s − 3.60i·7-s − 1.21·8-s + 1.73·9-s − 5.74i·11-s − 2.13i·12-s − 2.59·13-s − 1.12i·14-s + 3.42·16-s + (2.52 + 3.25i)17-s + 0.541·18-s + 4.28·19-s + ⋯
L(s)  = 1  + 0.219·2-s + 0.648i·3-s − 0.951·4-s + 0.142i·6-s − 1.36i·7-s − 0.429·8-s + 0.579·9-s − 1.73i·11-s − 0.616i·12-s − 0.718·13-s − 0.300i·14-s + 0.857·16-s + (0.612 + 0.790i)17-s + 0.127·18-s + 0.982·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03745 - 0.508569i\)
\(L(\frac12)\) \(\approx\) \(1.03745 - 0.508569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 - 3.25i)T \)
good2 \( 1 - 0.311T + 2T^{2} \)
3 \( 1 - 1.12iT - 3T^{2} \)
7 \( 1 + 3.60iT - 7T^{2} \)
11 \( 1 + 5.74iT - 11T^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
19 \( 1 - 4.28T + 19T^{2} \)
23 \( 1 + 1.47iT - 23T^{2} \)
29 \( 1 + 5.50iT - 29T^{2} \)
31 \( 1 + 8.33iT - 31T^{2} \)
37 \( 1 + 6.20iT - 37T^{2} \)
41 \( 1 - 3.95iT - 41T^{2} \)
43 \( 1 + 9.76T + 43T^{2} \)
47 \( 1 + 6.62T + 47T^{2} \)
53 \( 1 + 0.658T + 53T^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 - 8.76iT - 61T^{2} \)
67 \( 1 - 0.428T + 67T^{2} \)
71 \( 1 + 1.36iT - 71T^{2} \)
73 \( 1 + 3.25iT - 73T^{2} \)
79 \( 1 - 1.89iT - 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + 3.52T + 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−10.84783469923300946359005799336, −10.04262683519196172099114086464, −9.513295993431561715305642402357, −8.287624766115572797715725201302, −7.53080047813107415823293483667, −6.10567796359409548183083884162, −5.04807847161394215972179019271, −4.02929183418006490781308964749, −3.43853079081163864634765114323, −0.76891583825070497969614669581, 1.70902442509122564985231353401, 3.15255545924704785829957582422, 4.86019945901865684634734409210, 5.21430912355587137560393493930, 6.75757007331198180916139971413, 7.54520513245190358987985882696, 8.611546190382507213172758518339, 9.666650377934264430273116432521, 9.938487480016797730012846377589, 11.86164148955144113109740070244

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