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Properties

Label 2-475-19.16-c1-0-17
Degree $2$
Conductor $475$
Sign $0.580 + 0.814i$
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  + (1.36 + 1.14i)2-s + (−2.24 + 0.815i)3-s + (0.205 + 1.16i)4-s + (−3.99 − 1.45i)6-s + (−2.11 − 3.67i)7-s + (0.727 − 1.25i)8-s + (2.05 − 1.72i)9-s + (0.245 − 0.425i)11-s + (−1.41 − 2.44i)12-s + (−3.91 − 1.42i)13-s + (1.31 − 7.45i)14-s + (4.66 − 1.69i)16-s + (−1.55 − 1.30i)17-s + 4.78·18-s + (1.86 − 3.93i)19-s + ⋯
L(s)  = 1  + (0.966 + 0.811i)2-s + (−1.29 + 0.470i)3-s + (0.102 + 0.583i)4-s + (−1.63 − 0.594i)6-s + (−0.801 − 1.38i)7-s + (0.257 − 0.445i)8-s + (0.684 − 0.574i)9-s + (0.0740 − 0.128i)11-s + (−0.407 − 0.706i)12-s + (−1.08 − 0.394i)13-s + (0.351 − 1.99i)14-s + (1.16 − 0.424i)16-s + (−0.378 − 0.317i)17-s + 1.12·18-s + (0.428 − 0.903i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809507 - 0.417009i\)
\(L(\frac12)\) \(\approx\) \(0.809507 - 0.417009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 3.93i)T \)
good2 \( 1 + (-1.36 - 1.14i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (2.24 - 0.815i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.11 + 3.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.245 + 0.425i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.91 + 1.42i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.55 + 1.30i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.763 - 4.32i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.49 - 2.09i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.04 + 3.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.14T + 37T^{2} \)
41 \( 1 + (4.10 - 1.49i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.84 + 10.4i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.00 + 1.68i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.98 - 11.2i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.415 + 0.348i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.36 + 13.4i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.48 - 4.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.04 + 5.94i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.93 - 0.702i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.01 - 1.82i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.05 + 7.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.07 - 1.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.32 - 3.62i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.82558450857940218450494162316, −10.18616560857761955638460441162, −9.399051158871495437099738443555, −7.38244045643495165094266821325, −7.05840536431848700095553294008, −6.02637640362092736001619865287, −5.18376460206130316008226113705, −4.46443928772142364824399761995, −3.42922968319374840761934529146, −0.46332766107285340379460929942, 1.95165928719093413071880964368, 3.07796328817347492247024620029, 4.54004507455827464053215392669, 5.46709362951433286070321804529, 6.09080802964210349479743817026, 7.11520209331628169370976161453, 8.512075648518410425239003040034, 9.677345139718407260271218869663, 10.62297272738068966325213565878, 11.63307740350665805923171122108

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