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Properties

Label 2-425-17.13-c1-0-15
Degree $2$
Conductor $425$
Sign $0.990 + 0.135i$
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  − 1.21i·2-s + (2.23 + 2.23i)3-s + 0.520·4-s + (2.72 − 2.72i)6-s + (0.679 − 0.679i)7-s − 3.06i·8-s + 7.02i·9-s + (2.22 − 2.22i)11-s + (1.16 + 1.16i)12-s − 2.02·13-s + (−0.827 − 0.827i)14-s − 2.68·16-s + (−3.56 + 2.07i)17-s + 8.54·18-s − 5.28i·19-s + ⋯
L(s)  = 1  − 0.860i·2-s + (1.29 + 1.29i)3-s + 0.260·4-s + (1.11 − 1.11i)6-s + (0.256 − 0.256i)7-s − 1.08i·8-s + 2.34i·9-s + (0.669 − 0.669i)11-s + (0.336 + 0.336i)12-s − 0.561·13-s + (−0.221 − 0.221i)14-s − 0.672·16-s + (−0.864 + 0.503i)17-s + 2.01·18-s − 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33235 - 0.158610i\)
\(L(\frac12)\) \(\approx\) \(2.33235 - 0.158610i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.56 - 2.07i)T \)
good2 \( 1 + 1.21iT - 2T^{2} \)
3 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
7 \( 1 + (-0.679 + 0.679i)T - 7iT^{2} \)
11 \( 1 + (-2.22 + 2.22i)T - 11iT^{2} \)
13 \( 1 + 2.02T + 13T^{2} \)
19 \( 1 + 5.28iT - 19T^{2} \)
23 \( 1 + (6.01 - 6.01i)T - 23iT^{2} \)
29 \( 1 + (0.857 + 0.857i)T + 29iT^{2} \)
31 \( 1 + (-3.97 - 3.97i)T + 31iT^{2} \)
37 \( 1 + (-5.84 - 5.84i)T + 37iT^{2} \)
41 \( 1 + (1.04 - 1.04i)T - 41iT^{2} \)
43 \( 1 + 7.01iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 5.24iT - 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 + (-2.70 + 2.70i)T - 61iT^{2} \)
67 \( 1 - 2.37T + 67T^{2} \)
71 \( 1 + (-2.82 - 2.82i)T + 71iT^{2} \)
73 \( 1 + (5.51 + 5.51i)T + 73iT^{2} \)
79 \( 1 + (4.74 - 4.74i)T - 79iT^{2} \)
83 \( 1 + 0.171iT - 83T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 + (1.33 + 1.33i)T + 97iT^{2} \)
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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

−11.13101128591828150487779907317, −10.11974795929171807955062755274, −9.606700425158955655614470522545, −8.718924110166485278354653200752, −7.80538237376827397473721360277, −6.52985326865660435903331179001, −4.81730349072239867532030751190, −3.88957143844555709328203906435, −3.08320644606408303743453575178, −1.96549651939647515003145013441, 1.85047697566404541389391801234, 2.61787540288151183083181669698, 4.30429610288288223988786170786, 6.02610731776156825459808996457, 6.71583376797938924094302597142, 7.55338385465953895864185704091, 8.145483348427654656598428344334, 8.938797742862564669754741385977, 9.978020828988841633762228575980, 11.60159374028895044829164148975

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