Properties

Label 2-475-19.7-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.989 - 0.146i$
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  + (−0.155 − 0.269i)2-s + (−0.514 − 0.891i)3-s + (0.951 − 1.64i)4-s + (−0.160 + 0.277i)6-s − 3.28·7-s − 1.21·8-s + (0.969 − 1.67i)9-s − 5.16·11-s − 1.95·12-s + (−1.76 + 3.06i)13-s + (0.510 + 0.883i)14-s + (−1.71 − 2.96i)16-s + (0.504 + 0.874i)17-s − 0.603·18-s + (2.42 + 3.62i)19-s + ⋯
L(s)  = 1  + (−0.109 − 0.190i)2-s + (−0.297 − 0.514i)3-s + (0.475 − 0.824i)4-s + (−0.0653 + 0.113i)6-s − 1.23·7-s − 0.429·8-s + (0.323 − 0.559i)9-s − 1.55·11-s − 0.565·12-s + (−0.490 + 0.849i)13-s + (0.136 + 0.236i)14-s + (−0.428 − 0.742i)16-s + (0.122 + 0.211i)17-s − 0.142·18-s + (0.555 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.989 - 0.146i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0418423 + 0.568603i\)
\(L(\frac12)\) \(\approx\) \(0.0418423 + 0.568603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 - 3.62i)T \)
good2 \( 1 + (0.155 + 0.269i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.514 + 0.891i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 + (1.76 - 3.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.504 - 0.874i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.83 + 6.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 + 6.48T + 37T^{2} \)
41 \( 1 + (-3.40 - 5.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.15 + 5.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.92 + 3.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.55 + 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.06 - 5.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.59 + 9.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.227 - 0.394i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.06 + 3.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.44 + 2.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 + (-3.56 + 6.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.41 - 9.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.32809884221326706579902005774, −9.939863288178888537540994607591, −8.972735366119844676683989734950, −7.52413151649876329748431709850, −6.70663990490204204668848719098, −6.05934850837609855269038378923, −4.99020000725626619349971781804, −3.30063317785917401012725033145, −2.07592140050854673686409899301, −0.34063095852438283157953935224, 2.67928445108043370135981643861, 3.39235760829702096538997892795, 4.96475896142914929725905770808, 5.76155718095193681174931258220, 7.33084391420842670678441288904, 7.43502554301780282038583810703, 8.825565329562985984239285844326, 9.807825318758676312048628202564, 10.56113285119488086439643186392, 11.29533497100677146441135275292

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