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Properties

Label 2-475-19.9-c1-0-0
Degree $2$
Conductor $475$
Sign $0.0321 - 0.999i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (−0.370 − 2.09i)2-s + (−1.70 + 1.43i)3-s + (−2.39 + 0.871i)4-s + (3.64 + 3.05i)6-s + (0.742 − 1.28i)7-s + (0.583 + 1.01i)8-s + (0.342 − 1.94i)9-s + (−2.34 − 4.05i)11-s + (2.84 − 4.91i)12-s + (0.276 + 0.232i)13-s + (−2.97 − 1.08i)14-s + (−1.99 + 1.67i)16-s + (0.951 + 5.39i)17-s − 4.21·18-s + (−1.68 + 4.01i)19-s + ⋯
L(s)  = 1  + (−0.261 − 1.48i)2-s + (−0.986 + 0.827i)3-s + (−1.19 + 0.435i)4-s + (1.48 + 1.24i)6-s + (0.280 − 0.486i)7-s + (0.206 + 0.357i)8-s + (0.114 − 0.648i)9-s + (−0.705 − 1.22i)11-s + (0.819 − 1.42i)12-s + (0.0767 + 0.0643i)13-s + (−0.795 − 0.289i)14-s + (−0.499 + 0.418i)16-s + (0.230 + 1.30i)17-s − 0.992·18-s + (−0.386 + 0.922i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0321 - 0.999i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0321 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0700673 + 0.0678513i\)
\(L(\frac12)\) \(\approx\) \(0.0700673 + 0.0678513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (1.68 - 4.01i)T \)
good2 \( 1 + (0.370 + 2.09i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (1.70 - 1.43i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.742 + 1.28i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.34 + 4.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.276 - 0.232i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.951 - 5.39i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.79 - 2.10i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.155 - 0.882i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.40 - 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (4.01 - 3.36i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-6.78 - 2.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.88 + 10.7i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.12 - 2.23i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.70 + 9.65i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.20 - 0.803i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.53 - 8.71i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (6.02 + 2.19i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (2.19 - 1.83i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-1.58 + 1.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.08 + 5.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.54 + 2.13i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.819 + 4.64i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.97603954128064869871478421860, −10.44356631171530564274508308714, −10.15297473333344670263301637736, −8.821983065386961650928263269599, −7.970848052164423856368663691251, −6.24028541628553433368565711510, −5.38129938321566644807228773270, −4.12027795985301580235268045158, −3.42011976868083276085679369543, −1.67337901391075622524720056370, 0.07118817565899405906442109512, 2.26585644851765820000347716564, 4.70541321257788241465416514489, 5.42544145899488503925629814780, 6.24681670332983457771883691939, 7.19532296522056620594563428752, 7.55906623483935707392606322495, 8.722162670603851866094304644488, 9.634003153274637896359285640438, 10.84889196823314730601304071670

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