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Properties

Label 2-475-19.4-c1-0-4
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: $\chi_{475} (251, \cdot )$
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

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

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