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Properties

Label 2-475-19.5-c1-0-17
Degree $2$
Conductor $475$
Sign $0.336 + 0.941i$
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  + (−0.289 + 0.105i)2-s + (0.285 − 1.61i)3-s + (−1.45 + 1.22i)4-s + (0.0879 + 0.498i)6-s + (0.0445 + 0.0772i)7-s + (0.601 − 1.04i)8-s + (0.279 + 0.101i)9-s + (1.68 − 2.91i)11-s + (1.56 + 2.71i)12-s + (−0.0369 − 0.209i)13-s + (−0.0210 − 0.0176i)14-s + (0.597 − 3.38i)16-s + (−2.36 + 0.859i)17-s − 0.0916·18-s + (0.949 − 4.25i)19-s + ⋯
L(s)  = 1  + (−0.204 + 0.0744i)2-s + (0.164 − 0.934i)3-s + (−0.729 + 0.612i)4-s + (0.0358 + 0.203i)6-s + (0.0168 + 0.0291i)7-s + (0.212 − 0.368i)8-s + (0.0931 + 0.0339i)9-s + (0.507 − 0.879i)11-s + (0.452 + 0.782i)12-s + (−0.0102 − 0.0581i)13-s + (−0.00562 − 0.00471i)14-s + (0.149 − 0.846i)16-s + (−0.573 + 0.208i)17-s − 0.0215·18-s + (0.217 − 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922183 - 0.649930i\)
\(L(\frac12)\) \(\approx\) \(0.922183 - 0.649930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.949 + 4.25i)T \)
good2 \( 1 + (0.289 - 0.105i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.285 + 1.61i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.0445 - 0.0772i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.68 + 2.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0369 + 0.209i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.36 - 0.859i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.57 + 3.83i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.51 - 1.64i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.03 + 6.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.84T + 37T^{2} \)
41 \( 1 + (0.523 - 2.96i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.87 - 1.57i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (7.15 + 2.60i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.43 + 5.39i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (9.80 - 3.56i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.757 - 0.635i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-9.37 - 3.41i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.73 - 3.97i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.73 - 15.5i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.178 - 1.01i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.96 - 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.113 - 0.646i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-15.7 + 5.75i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.94922519875637326366852409699, −9.722505666825080236479968684223, −8.761888168106627780033467936672, −8.251246990375135113064974630432, −7.18251702931147541864457600205, −6.52767282102470433686270079837, −5.05258633743337494282726242390, −3.92761958588758338084428090226, −2.61878131100051313217476053359, −0.833391530888390168434921862557, 1.54839575071696143526044097957, 3.51437240029836391760238988004, 4.51279748539710770735612700440, 5.17768865123847055910801139987, 6.50396623955979514538136065981, 7.65963192358555279780430215311, 9.057176012980262469158113933561, 9.265818995092937591588389085390, 10.26382055873754419100645679953, 10.73915280611904096315670582118

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