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Properties

Label 2-475-95.49-c1-0-13
Degree $2$
Conductor $475$
Sign $-0.570 + 0.821i$
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  + (−2.38 − 1.37i)2-s + (1.29 + 0.745i)3-s + (2.80 + 4.85i)4-s + (−2.05 − 3.56i)6-s − 2.84i·7-s − 9.94i·8-s + (−0.387 − 0.670i)9-s − 0.864·11-s + 8.36i·12-s + (−0.557 + 0.321i)13-s + (−3.92 + 6.80i)14-s + (−8.11 + 14.0i)16-s + (−3.24 − 1.87i)17-s + 2.13i·18-s + (3.36 − 2.77i)19-s + ⋯
L(s)  = 1  + (−1.68 − 0.975i)2-s + (0.745 + 0.430i)3-s + (1.40 + 2.42i)4-s + (−0.839 − 1.45i)6-s − 1.07i·7-s − 3.51i·8-s + (−0.129 − 0.223i)9-s − 0.260·11-s + 2.41i·12-s + (−0.154 + 0.0892i)13-s + (−1.04 + 1.81i)14-s + (−2.02 + 3.51i)16-s + (−0.785 − 0.453i)17-s + 0.503i·18-s + (0.770 − 0.636i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.570 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296477 - 0.567264i\)
\(L(\frac12)\) \(\approx\) \(0.296477 - 0.567264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.36 + 2.77i)T \)
good2 \( 1 + (2.38 + 1.37i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.29 - 0.745i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.84iT - 7T^{2} \)
11 \( 1 + 0.864T + 11T^{2} \)
13 \( 1 + (0.557 - 0.321i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.24 + 1.87i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.361 - 0.208i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.85 + 8.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.93T + 31T^{2} \)
37 \( 1 - 6.36iT - 37T^{2} \)
41 \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.78 + 1.02i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.42 - 1.97i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.51 + 5.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.22 + 2.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 - 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.891 + 1.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.17 + 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.912 + 1.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.43iT - 83T^{2} \)
89 \( 1 + (-2.22 - 3.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.39 - 5.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.39617358558140195200183405123, −9.752827463111885489699354845062, −9.158844243089965356423597401771, −8.248760702837659772933984814448, −7.51893569233095065475804367427, −6.62872146863758114900020991383, −4.29450825906059274415789143792, −3.32492006724431094756050702484, −2.32622309478224459092833906682, −0.60758512182353977284587055371, 1.68472625871532679793876505187, 2.68598670088307410929299433322, 5.25294048776519998818115401606, 6.01149065383011164621511637040, 7.16808864961212733606792204583, 7.82107749728842002161402845556, 8.729439844004911969768029206366, 9.003595908506411624766289029335, 10.10081840994109051076733700695, 10.94475416219501334751929460361

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