Properties

Label 2-475-95.64-c1-0-5
Degree $2$
Conductor $475$
Sign $-0.884 - 0.466i$
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  + (−1.97 + 1.14i)2-s + (−2.17 + 1.25i)3-s + (1.61 − 2.79i)4-s + (2.86 − 4.96i)6-s − 3.50i·7-s + 2.79i·8-s + (1.64 − 2.84i)9-s − 4.50·11-s + 8.07i·12-s + (4.33 + 2.5i)13-s + (4.00 + 6.94i)14-s + (0.0316 + 0.0547i)16-s + (−0.137 + 0.0793i)17-s + 7.50i·18-s + (4.26 + 0.920i)19-s + ⋯
L(s)  = 1  + (−1.39 + 0.807i)2-s + (−1.25 + 0.723i)3-s + (0.805 − 1.39i)4-s + (1.16 − 2.02i)6-s − 1.32i·7-s + 0.987i·8-s + (0.547 − 0.948i)9-s − 1.35·11-s + 2.33i·12-s + (1.20 + 0.693i)13-s + (1.07 + 1.85i)14-s + (0.00790 + 0.0136i)16-s + (−0.0333 + 0.0192i)17-s + 1.76i·18-s + (0.977 + 0.211i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.884 - 0.466i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0650988 + 0.262795i\)
\(L(\frac12)\) \(\approx\) \(0.0650988 + 0.262795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.26 - 0.920i)T \)
good2 \( 1 + (1.97 - 1.14i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.17 - 1.25i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.50iT - 7T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + (-4.33 - 2.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.137 - 0.0793i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.00 + 0.579i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 + (3.03 + 5.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.89 - 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.65 - 1.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.97 - 2.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.53 - 2.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.436 + 0.756i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.31 + 4.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.11 - 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.19 - 3.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.06 + 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.85iT - 83T^{2} \)
89 \( 1 + (0.556 - 0.963i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.40 - 0.809i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.87280672682545611044340277150, −10.39187353277291650432306637159, −9.836408707394721748854110969710, −8.677106249654338087762374588911, −7.70375836801689492122485772897, −6.90965040752628556359587439651, −5.99963363457053245665093239141, −5.05139435061930171573947220859, −3.80741101720927040588792293694, −1.08429278374037635509062596045, 0.37708249288552300273319581125, 1.85252958895028047670747355571, 3.04078399755626853139889848622, 5.41514099913384297507747464977, 5.79759106603790936472737967869, 7.23717958361812601533207317094, 8.076908395850461028244962850638, 8.868537697808641011754112803265, 9.890149042148019190927219641859, 10.83544308951075331680898620611

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