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Properties

Label 2-475-19.6-c1-0-23
Degree $2$
Conductor $475$
Sign $-0.402 + 0.915i$
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  + (1.24 − 1.04i)2-s + (−1.11 − 0.406i)3-s + (0.107 − 0.611i)4-s + (−1.80 + 0.658i)6-s + (1.11 − 1.93i)7-s + (1.11 + 1.93i)8-s + (−1.21 − 1.02i)9-s + (−2.82 − 4.88i)11-s + (−0.369 + 0.639i)12-s + (4.41 − 1.60i)13-s + (−0.627 − 3.55i)14-s + (4.56 + 1.66i)16-s + (0.601 − 0.505i)17-s − 2.56·18-s + (−2.63 − 3.47i)19-s + ⋯
L(s)  = 1  + (0.877 − 0.735i)2-s + (−0.644 − 0.234i)3-s + (0.0539 − 0.305i)4-s + (−0.738 + 0.268i)6-s + (0.421 − 0.730i)7-s + (0.394 + 0.683i)8-s + (−0.405 − 0.340i)9-s + (−0.850 − 1.47i)11-s + (−0.106 + 0.184i)12-s + (1.22 − 0.445i)13-s + (−0.167 − 0.951i)14-s + (1.14 + 0.415i)16-s + (0.146 − 0.122i)17-s − 0.605·18-s + (−0.603 − 0.797i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.402 + 0.915i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936230 - 1.43493i\)
\(L(\frac12)\) \(\approx\) \(0.936230 - 1.43493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 3.47i)T \)
good2 \( 1 + (-1.24 + 1.04i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (1.11 + 0.406i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.11 + 1.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.82 + 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.41 + 1.60i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.601 + 0.505i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.890 + 5.05i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.32 - 1.95i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.05 - 7.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.985T + 37T^{2} \)
41 \( 1 + (1.33 + 0.484i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.264 - 1.49i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.735 + 0.617i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.766 - 4.34i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-7.56 + 6.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.363 + 2.06i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.48 - 2.08i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.24 - 7.06i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-14.2 - 5.18i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.25 + 0.458i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (3.76 - 6.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.19 - 1.16i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-9.77 + 8.20i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.89347727115916676872548763265, −10.62604842686360597290906673495, −8.677248635781558647046649457784, −8.194079680433492323784792041691, −6.77128477631350869762131816930, −5.74208781234073788774106614870, −4.96335497901069045493254732790, −3.70792852528774303260998812250, −2.84156006611253946557338939168, −0.908776451496382630385220995431, 2.01130072139356479160717490692, 3.90602635785152016867137761994, 4.89653743006063284202802245021, 5.59261961204836585719194808507, 6.27653056161391134727193114432, 7.46438257830505592668517243400, 8.340020306992179327239220098941, 9.644623581103711445197235086505, 10.48785100349819829362721521899, 11.38423684095686788322120867962

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