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Properties

Label 2-425-17.9-c1-0-9
Degree $2$
Conductor $425$
Sign $-0.675 - 0.737i$
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  + (−1.82 + 1.82i)2-s + (2.84 + 1.17i)3-s − 4.67i·4-s + (−7.35 + 3.04i)6-s + (1.07 + 2.60i)7-s + (4.89 + 4.89i)8-s + (4.59 + 4.59i)9-s + (0.616 − 0.255i)11-s + (5.51 − 13.3i)12-s − 4.34i·13-s + (−6.72 − 2.78i)14-s − 8.52·16-s + (2.64 + 3.16i)17-s − 16.7·18-s + (1.88 − 1.88i)19-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)2-s + (1.64 + 0.680i)3-s − 2.33i·4-s + (−3.00 + 1.24i)6-s + (0.407 + 0.983i)7-s + (1.72 + 1.72i)8-s + (1.53 + 1.53i)9-s + (0.185 − 0.0769i)11-s + (1.59 − 3.84i)12-s − 1.20i·13-s + (−1.79 − 0.744i)14-s − 2.13·16-s + (0.640 + 0.767i)17-s − 3.95·18-s + (0.433 − 0.433i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.675 - 0.737i$
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: \((2,\ 425,\ (\ :1/2),\ -0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528654 + 1.20145i\)
\(L(\frac12)\) \(\approx\) \(0.528654 + 1.20145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 3.16i)T \)
good2 \( 1 + (1.82 - 1.82i)T - 2iT^{2} \)
3 \( 1 + (-2.84 - 1.17i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.07 - 2.60i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.616 + 0.255i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 4.34iT - 13T^{2} \)
19 \( 1 + (-1.88 + 1.88i)T - 19iT^{2} \)
23 \( 1 + (2.78 - 1.15i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.50 - 3.63i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (6.05 + 2.50i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.82 + 1.16i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.38 + 5.76i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.37 - 4.37i)T + 43iT^{2} \)
47 \( 1 - 1.08iT - 47T^{2} \)
53 \( 1 + (4.94 - 4.94i)T - 53iT^{2} \)
59 \( 1 + (0.272 + 0.272i)T + 59iT^{2} \)
61 \( 1 + (4.24 + 10.2i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (-4.19 - 1.73i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.08 + 7.43i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-6.31 + 2.61i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 - 0.844iT - 89T^{2} \)
97 \( 1 + (-4.46 + 10.7i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.84533484855384456090740744846, −10.11448339685003879497124021618, −9.203216738726132978379532524091, −8.836914821775752174432864865199, −7.912799199672709807933022699425, −7.55914332236753176383529720557, −5.95762309619458057423200703943, −5.06393877440666424331129085796, −3.37574913659667487736230622975, −1.87355188280708928900499971725, 1.24989077912236684490144122160, 2.13543984578405140712748433605, 3.34797909526048758706682886503, 4.15014978202593474947189452783, 6.97042946762103618964944383065, 7.59081766173963385650954927334, 8.260832402779264327490635857877, 9.205484833912586819301868855535, 9.668702122519307586690927687184, 10.62966853671169717802831385838

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