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Properties

Label 2-475-19.4-c1-0-0
Degree $2$
Conductor $475$
Sign $-0.999 - 0.0359i$
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  + (0.234 + 0.0854i)2-s + (0.399 + 2.26i)3-s + (−1.48 − 1.24i)4-s + (−0.0998 + 0.566i)6-s + (−1.98 + 3.42i)7-s + (−0.491 − 0.851i)8-s + (−2.15 + 0.785i)9-s + (−1.56 − 2.71i)11-s + (2.22 − 3.86i)12-s + (−0.461 + 2.61i)13-s + (−0.757 + 0.635i)14-s + (0.630 + 3.57i)16-s + (−2.12 − 0.771i)17-s − 0.573·18-s + (−3.76 + 2.19i)19-s + ⋯
L(s)  = 1  + (0.165 + 0.0604i)2-s + (0.230 + 1.30i)3-s + (−0.742 − 0.622i)4-s + (−0.0407 + 0.231i)6-s + (−0.748 + 1.29i)7-s + (−0.173 − 0.301i)8-s + (−0.719 + 0.261i)9-s + (−0.472 − 0.818i)11-s + (0.643 − 1.11i)12-s + (−0.128 + 0.726i)13-s + (−0.202 + 0.169i)14-s + (0.157 + 0.893i)16-s + (−0.514 − 0.187i)17-s − 0.135·18-s + (−0.864 + 0.502i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.999 - 0.0359i$
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: \((2,\ 475,\ (\ :1/2),\ -0.999 - 0.0359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0115253 + 0.641112i\)
\(L(\frac12)\) \(\approx\) \(0.0115253 + 0.641112i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.76 - 2.19i)T \)
good2 \( 1 + (-0.234 - 0.0854i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.399 - 2.26i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (1.98 - 3.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.56 + 2.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.461 - 2.61i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.12 + 0.771i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.81 + 4.87i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.28 - 0.466i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.447 - 0.774i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.62T + 37T^{2} \)
41 \( 1 + (-1.08 - 6.14i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.35 - 1.13i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.165 - 0.0603i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.19 - 4.35i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-7.34 - 2.67i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.796 + 0.668i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (14.2 - 5.20i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-7.99 + 6.70i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.726 - 4.11i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.94 - 11.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (6.62 - 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.257 + 1.46i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-14.4 - 5.26i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.21801670869433714149839130871, −10.28354403256449482437957195674, −9.663368880683624033285540188773, −8.923983198697085041886394684418, −8.392487633880497989775771388590, −6.36777607323006830575080670667, −5.73660843887175051602297518481, −4.67046107569152490355814265356, −3.86244340413052539429879726826, −2.54247472137194479976525140267, 0.34902182993975605244448970177, 2.23927786278971267086524421268, 3.59653696698039412900954200364, 4.57342958705395631668537867257, 6.07452079215056261001593424337, 7.24498599235944499806536028598, 7.57436306151681728967524979721, 8.510273084773165747541398152821, 9.702354214663381261678245654619, 10.46415098783832464713778370129

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