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Properties

Label 2-475-19.4-c1-0-13
Degree $2$
Conductor $475$
Sign $-0.349 - 0.936i$
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  + (2.22 + 0.810i)2-s + (0.396 + 2.25i)3-s + (2.77 + 2.32i)4-s + (−0.940 + 5.33i)6-s + (−0.818 + 1.41i)7-s + (1.92 + 3.32i)8-s + (−2.08 + 0.759i)9-s + (−1.36 − 2.36i)11-s + (−4.13 + 7.16i)12-s + (1.07 − 6.10i)13-s + (−2.97 + 2.49i)14-s + (0.325 + 1.84i)16-s + (2.93 + 1.06i)17-s − 5.26·18-s + (−3.46 − 2.64i)19-s + ⋯
L(s)  = 1  + (1.57 + 0.573i)2-s + (0.229 + 1.29i)3-s + (1.38 + 1.16i)4-s + (−0.384 + 2.17i)6-s + (−0.309 + 0.535i)7-s + (0.679 + 1.17i)8-s + (−0.695 + 0.253i)9-s + (−0.411 − 0.712i)11-s + (−1.19 + 2.06i)12-s + (0.298 − 1.69i)13-s + (−0.794 + 0.666i)14-s + (0.0812 + 0.460i)16-s + (0.712 + 0.259i)17-s − 1.24·18-s + (−0.795 − 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.349 - 0.936i$
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: \((2,\ 475,\ (\ :1/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93766 + 2.79237i\)
\(L(\frac12)\) \(\approx\) \(1.93766 + 2.79237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 + 2.64i)T \)
good2 \( 1 + (-2.22 - 0.810i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.396 - 2.25i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.818 - 1.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.07 + 6.10i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.93 - 1.06i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.59 - 4.69i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.09 - 0.761i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.21 - 3.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.04T + 37T^{2} \)
41 \( 1 + (-0.681 - 3.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.362 + 0.303i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.16 - 0.787i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.87 + 4.09i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (11.5 + 4.19i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.66 + 3.07i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.630 + 0.229i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.01 + 1.69i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.13 + 6.41i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.715 - 4.05i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.21 - 5.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.00 + 17.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.122 - 0.0444i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.22029012649334037900868738302, −10.62790814334303883226804110733, −9.531460354515942747089811451730, −8.530486090451469431175563465284, −7.47611006372346065042759143410, −6.12408043824884504180697983303, −5.40953458230437230179038816335, −4.74389769730022921744122079530, −3.28146520556720065727937604446, −3.19427803363984555473572299719, 1.59385646332243431840060803755, 2.51053171031841980427214318222, 3.86599924063429732878169494983, 4.75840290816225191662735037669, 6.11393590633264392515207177180, 6.81085712327506117191804618948, 7.55048876428364678255815511875, 8.904087816781988585819576230762, 10.21607670906041079364137953940, 11.12909738713597008099779731315

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