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Properties

Label 2-475-19.17-c1-0-17
Degree $2$
Conductor $475$
Sign $-0.441 + 0.897i$
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  + (0.253 − 1.44i)2-s + (−1.15 − 0.970i)3-s + (−0.130 − 0.0473i)4-s + (−1.69 + 1.41i)6-s + (2.03 + 3.52i)7-s + (1.36 − 2.35i)8-s + (−0.124 − 0.707i)9-s + (0.310 − 0.537i)11-s + (0.104 + 0.180i)12-s + (3.90 − 3.27i)13-s + (5.59 − 2.03i)14-s + (−3.26 − 2.73i)16-s + (0.0462 − 0.262i)17-s − 1.05·18-s + (0.399 − 4.34i)19-s + ⋯
L(s)  = 1  + (0.179 − 1.01i)2-s + (−0.667 − 0.560i)3-s + (−0.0650 − 0.0236i)4-s + (−0.690 + 0.579i)6-s + (0.769 + 1.33i)7-s + (0.481 − 0.833i)8-s + (−0.0416 − 0.235i)9-s + (0.0936 − 0.162i)11-s + (0.0301 + 0.0522i)12-s + (1.08 − 0.908i)13-s + (1.49 − 0.544i)14-s + (−0.815 − 0.684i)16-s + (0.0112 − 0.0636i)17-s − 0.247·18-s + (0.0917 − 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806882 - 1.29556i\)
\(L(\frac12)\) \(\approx\) \(0.806882 - 1.29556i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.399 + 4.34i)T \)
good2 \( 1 + (-0.253 + 1.44i)T + (-1.87 - 0.684i)T^{2} \)
3 \( 1 + (1.15 + 0.970i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.03 - 3.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.310 + 0.537i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.90 + 3.27i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.0462 + 0.262i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (5.48 + 1.99i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.708 - 4.01i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.24 - 5.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.83T + 37T^{2} \)
41 \( 1 + (3.43 + 2.88i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.69 - 0.615i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.00 + 11.3i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.37 + 0.862i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.154 + 0.876i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.03 + 0.742i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.44 - 13.8i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.54 + 0.563i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (1.37 + 1.15i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.94 - 3.30i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.89 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.000572 - 0.000480i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-1.80 + 10.2i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.13537996624250194585558962664, −10.17151962696093267307347579609, −8.943541120129054638400124798427, −8.157986899541695023883079513273, −6.83273263405043331131921440985, −5.97957523299555604749898213010, −5.01558772094672058835113507960, −3.51220296086303505427511386873, −2.37032538451741711355633480771, −1.09050703385577844778739219120, 1.69354631658320615048801074865, 4.10677387857238847106870005266, 4.59828631302197015383105260391, 5.87273674702709286340507311844, 6.44071214359331156877990616973, 7.83750625949449203820397495947, 7.972971704984196060688609453392, 9.658977124342144041606733549641, 10.55787409350004057009744431155, 11.21878009467011147864849437951

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