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Properties

Label 2-475-19.6-c1-0-1
Degree $2$
Conductor $475$
Sign $-0.671 - 0.741i$
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  + (0.454 − 0.381i)2-s + (−1.81 − 0.660i)3-s + (−0.286 + 1.62i)4-s + (−1.07 + 0.392i)6-s + (0.530 − 0.918i)7-s + (1.08 + 1.87i)8-s + (0.555 + 0.466i)9-s + (−0.0983 − 0.170i)11-s + (1.58 − 2.75i)12-s + (−4.96 + 1.80i)13-s + (−0.109 − 0.620i)14-s + (−1.88 − 0.686i)16-s + (−0.540 + 0.453i)17-s + 0.430·18-s + (−4.24 + 0.983i)19-s + ⋯
L(s)  = 1  + (0.321 − 0.269i)2-s + (−1.04 − 0.381i)3-s + (−0.143 + 0.811i)4-s + (−0.439 + 0.160i)6-s + (0.200 − 0.347i)7-s + (0.382 + 0.663i)8-s + (0.185 + 0.155i)9-s + (−0.0296 − 0.0513i)11-s + (0.458 − 0.794i)12-s + (−1.37 + 0.501i)13-s + (−0.0292 − 0.165i)14-s + (−0.471 − 0.171i)16-s + (−0.130 + 0.109i)17-s + 0.101·18-s + (−0.974 + 0.225i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.671 - 0.741i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.671 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152394 + 0.343684i\)
\(L(\frac12)\) \(\approx\) \(0.152394 + 0.343684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.24 - 0.983i)T \)
good2 \( 1 + (-0.454 + 0.381i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (1.81 + 0.660i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.530 + 0.918i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0983 + 0.170i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.96 - 1.80i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.540 - 0.453i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.15 - 6.52i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.59 + 2.17i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.95 - 6.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.09T + 37T^{2} \)
41 \( 1 + (-1.27 - 0.463i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.56 + 8.87i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.69 + 3.09i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.924 + 5.24i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (8.41 - 7.05i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.04 + 11.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.34 - 1.96i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.434 - 2.46i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-6.15 - 2.24i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-11.5 - 4.19i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (2.01 - 3.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (16.4 - 5.99i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-1.02 + 0.861i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.44293369117273289068807805280, −10.84990039724212590348127838897, −9.653355836540408734476829856275, −8.556796261253809091636933952408, −7.46699544000035772188883748788, −6.83912871278679353670396283790, −5.55128114530389933429084927231, −4.68387996798714908970544520434, −3.56857902928932416554745861054, −2.01785663913652846496108970188, 0.22120811343520442404390576666, 2.33347306263074409057145418157, 4.39880698775792811546485959967, 4.99361792761047576456330213247, 5.85127167658182720176542092452, 6.59727672105720596387592619785, 7.80488191349702700308200080082, 9.104697706288602135866913810973, 10.02557077010154604901193820453, 10.67793655753114891438130017580

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