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Properties

Label 2-475-19.9-c1-0-25
Degree $2$
Conductor $475$
Sign $-0.218 + 0.975i$
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  + (−0.124 − 0.708i)2-s + (0.945 − 0.793i)3-s + (1.39 − 0.507i)4-s + (−0.680 − 0.570i)6-s + (0.645 − 1.11i)7-s + (−1.25 − 2.16i)8-s + (−0.256 + 1.45i)9-s + (−2.88 − 4.99i)11-s + (0.914 − 1.58i)12-s + (−1.80 − 1.51i)13-s + (−0.873 − 0.317i)14-s + (0.890 − 0.747i)16-s + (1.18 + 6.74i)17-s + 1.06·18-s + (2.40 − 3.63i)19-s + ⋯
L(s)  = 1  + (−0.0883 − 0.501i)2-s + (0.545 − 0.458i)3-s + (0.696 − 0.253i)4-s + (−0.277 − 0.233i)6-s + (0.244 − 0.422i)7-s + (−0.442 − 0.767i)8-s + (−0.0854 + 0.484i)9-s + (−0.869 − 1.50i)11-s + (0.264 − 0.457i)12-s + (−0.499 − 0.419i)13-s + (−0.233 − 0.0849i)14-s + (0.222 − 0.186i)16-s + (0.288 + 1.63i)17-s + 0.250·18-s + (0.551 − 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13752 - 1.41973i\)
\(L(\frac12)\) \(\approx\) \(1.13752 - 1.41973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.40 + 3.63i)T \)
good2 \( 1 + (0.124 + 0.708i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.945 + 0.793i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.645 + 1.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.88 + 4.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.80 + 1.51i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.18 - 6.74i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-5.32 + 1.93i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.04 - 5.92i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.19 + 2.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + (3.51 - 2.94i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.260 - 0.0947i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.0880 - 0.499i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.75 - 2.45i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.00 - 5.67i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.77 + 2.46i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.77 + 10.0i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.05 - 0.382i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.83 - 1.53i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (10.7 - 9.05i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.608 + 1.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.85 + 5.74i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.83 - 10.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.87766943231833211156177520919, −10.17939794295613407405282690717, −8.847214804966043493695481405111, −8.005198457585066072930536371350, −7.27514317104108061293726244473, −6.12089401798564060333114108627, −5.10020267520860890105495670311, −3.32435863581037559124276614784, −2.58503724344085286794750534020, −1.14341178061893620179064431039, 2.23035657582304366768926290519, 3.13557116969080116653558464551, 4.67500065614795419971120434757, 5.59450435271631669374658353911, 6.97822927698599317702663826196, 7.49030236296050061178468351571, 8.481896635794222410500191570118, 9.545684114739642754159141421799, 10.02383923020512664048906019929, 11.48720294901338311562054031579

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