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Properties

Label 2-475-19.9-c1-0-4
Degree $2$
Conductor $475$
Sign $-0.754 + 0.655i$
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  + (0.256 + 1.45i)2-s + (−1.90 + 1.59i)3-s + (−0.176 + 0.0642i)4-s + (−2.81 − 2.36i)6-s + (−1.62 + 2.81i)7-s + (1.34 + 2.32i)8-s + (0.549 − 3.11i)9-s + (2.09 + 3.62i)11-s + (0.233 − 0.404i)12-s + (−1.36 − 1.14i)13-s + (−4.51 − 1.64i)14-s + (−3.32 + 2.79i)16-s + (−1.09 − 6.23i)17-s + 4.68·18-s + (−4.09 + 1.49i)19-s + ⋯
L(s)  = 1  + (0.181 + 1.03i)2-s + (−1.09 + 0.921i)3-s + (−0.0883 + 0.0321i)4-s + (−1.14 − 0.963i)6-s + (−0.614 + 1.06i)7-s + (0.473 + 0.820i)8-s + (0.183 − 1.03i)9-s + (0.630 + 1.09i)11-s + (0.0673 − 0.116i)12-s + (−0.379 − 0.318i)13-s + (−1.20 − 0.439i)14-s + (−0.831 + 0.697i)16-s + (−0.266 − 1.51i)17-s + 1.10·18-s + (−0.939 + 0.343i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.754 + 0.655i$
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: \((2,\ 475,\ (\ :1/2),\ -0.754 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305369 - 0.817104i\)
\(L(\frac12)\) \(\approx\) \(0.305369 - 0.817104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.09 - 1.49i)T \)
good2 \( 1 + (-0.256 - 1.45i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (1.90 - 1.59i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.62 - 2.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.36 + 1.14i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.09 + 6.23i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.34 + 0.490i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0589 - 0.334i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.38 + 2.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 + (5.46 - 4.58i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-9.29 - 3.38i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.0773 + 0.438i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.80 - 2.47i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.545 - 3.09i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.88 + 1.04i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.48 - 8.42i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.1 + 4.40i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.39 + 1.17i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-0.535 + 0.449i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.276 - 0.478i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.23 - 4.38i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.29 - 13.0i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.61959291289822366472055214564, −10.64822073181035259062852737122, −9.715216232161253516724477384211, −9.031366698606207119285347609622, −7.64778568538761008123688260118, −6.61527539612180144333972912608, −5.99559483585476874489179348264, −5.07494721419007775086996353924, −4.42243480389568412514045184106, −2.49811669438537171770494386780, 0.56501364424370510695096880535, 1.76886159732232086848747463334, 3.41914842007304229920260825270, 4.35456032111484403884125375994, 6.03194763472572511488461704603, 6.63479919822524369483001881620, 7.37356402411549460510275887691, 8.752875328823683178284935853272, 10.08698898190203697579553745773, 10.83719170197101450599713158015

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