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Properties

Label 2-425-17.13-c1-0-16
Degree $2$
Conductor $425$
Sign $0.941 + 0.337i$
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  − 0.0601i·2-s + (0.294 + 0.294i)3-s + 1.99·4-s + (0.0177 − 0.0177i)6-s + (0.900 − 0.900i)7-s − 0.240i·8-s − 2.82i·9-s + (2.24 − 2.24i)11-s + (0.588 + 0.588i)12-s − 4.23·13-s + (−0.0542 − 0.0542i)14-s + 3.97·16-s + (3.91 − 1.29i)17-s − 0.170·18-s + 4.76i·19-s + ⋯
L(s)  = 1  − 0.0425i·2-s + (0.170 + 0.170i)3-s + 0.998·4-s + (0.00724 − 0.00724i)6-s + (0.340 − 0.340i)7-s − 0.0850i·8-s − 0.942i·9-s + (0.677 − 0.677i)11-s + (0.169 + 0.169i)12-s − 1.17·13-s + (−0.0144 − 0.0144i)14-s + 0.994·16-s + (0.949 − 0.313i)17-s − 0.0400·18-s + 1.09i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.941 + 0.337i$
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: \((2,\ 425,\ (\ :1/2),\ 0.941 + 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84185 - 0.319736i\)
\(L(\frac12)\) \(\approx\) \(1.84185 - 0.319736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.91 + 1.29i)T \)
good2 \( 1 + 0.0601iT - 2T^{2} \)
3 \( 1 + (-0.294 - 0.294i)T + 3iT^{2} \)
7 \( 1 + (-0.900 + 0.900i)T - 7iT^{2} \)
11 \( 1 + (-2.24 + 2.24i)T - 11iT^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
19 \( 1 - 4.76iT - 19T^{2} \)
23 \( 1 + (5.13 - 5.13i)T - 23iT^{2} \)
29 \( 1 + (1.35 + 1.35i)T + 29iT^{2} \)
31 \( 1 + (-1.64 - 1.64i)T + 31iT^{2} \)
37 \( 1 + (-3.84 - 3.84i)T + 37iT^{2} \)
41 \( 1 + (-0.0814 + 0.0814i)T - 41iT^{2} \)
43 \( 1 - 0.562iT - 43T^{2} \)
47 \( 1 + 5.01T + 47T^{2} \)
53 \( 1 + 7.75iT - 53T^{2} \)
59 \( 1 + 2.01iT - 59T^{2} \)
61 \( 1 + (7.03 - 7.03i)T - 61iT^{2} \)
67 \( 1 - 3.64T + 67T^{2} \)
71 \( 1 + (6.92 + 6.92i)T + 71iT^{2} \)
73 \( 1 + (-10.2 - 10.2i)T + 73iT^{2} \)
79 \( 1 + (5.78 - 5.78i)T - 79iT^{2} \)
83 \( 1 + 4.38iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (9.15 + 9.15i)T + 97iT^{2} \)
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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

−11.32986733176788298697892277025, −10.04955783534204291734822977002, −9.644628272226337983134911359538, −8.199654871265139544285511764641, −7.46178370584592856554485533023, −6.42932378570121808438756028725, −5.57065346872124930769887191542, −3.96989844589776574335558777433, −3.02745692945773412426452540796, −1.41288717908302102739439640599, 1.86495210567774231037614642957, 2.72580324467043436304875022271, 4.46026042414188366636197438833, 5.53315752270567903395388539111, 6.71023470601012865070337423406, 7.51862361379477437447196857247, 8.230767151864651384742334845540, 9.548380199098634830398241815366, 10.38891501941299523394357118898, 11.27131366610380760935876619747

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