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Properties

Label 2-425-85.84-c1-0-8
Degree $2$
Conductor $425$
Sign $0.869 + 0.493i$
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  − 0.311i·2-s − 2.21·3-s + 1.90·4-s + 0.688i·6-s + 1.59·7-s − 1.21i·8-s + 1.90·9-s + 1.31i·11-s − 4.21·12-s − 3.52i·13-s − 0.495i·14-s + 3.42·16-s + (−0.214 + 4.11i)17-s − 0.592i·18-s + 4.42·19-s + ⋯
L(s)  = 1  − 0.219i·2-s − 1.27·3-s + 0.951·4-s + 0.281i·6-s + 0.601·7-s − 0.429i·8-s + 0.634·9-s + 0.395i·11-s − 1.21·12-s − 0.977i·13-s − 0.132i·14-s + 0.857·16-s + (−0.0519 + 0.998i)17-s − 0.139i·18-s + 1.01·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.869 + 0.493i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19078 - 0.314004i\)
\(L(\frac12)\) \(\approx\) \(1.19078 - 0.314004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.214 - 4.11i)T \)
good2 \( 1 + 0.311iT - 2T^{2} \)
3 \( 1 + 2.21T + 3T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
11 \( 1 - 1.31iT - 11T^{2} \)
13 \( 1 + 3.52iT - 13T^{2} \)
19 \( 1 - 4.42T + 19T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 + 8.42iT - 29T^{2} \)
31 \( 1 + 7.73iT - 31T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
41 \( 1 - 3.67iT - 41T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 - 3.33iT - 47T^{2} \)
53 \( 1 - 9.18iT - 53T^{2} \)
59 \( 1 + 1.37T + 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + 9.13iT - 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 + 7.87iT - 79T^{2} \)
83 \( 1 + 7.19iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−11.22596589667978667591091897077, −10.52519131473119589888137858327, −9.690424878640263606601394998425, −8.069075801465377201292977222979, −7.33698265428148242075495981078, −6.13647168125816015178390361533, −5.61845912067730794504720761322, −4.36935134997011436598073242324, −2.74962235061432912699463621605, −1.13603404189222838774170991547, 1.34243052612499602607037004279, 3.06264805603479819664731484306, 4.91349882651599568858372072230, 5.46466363181753619514059721198, 6.70075148611286889831039783285, 7.10382098999614363434900415178, 8.396204287313037942375957748845, 9.573784611108197134163864229208, 10.86821560144955543357181498409, 11.21526428895231987991920206822

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