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Properties

Label 2-475-95.12-c1-0-8
Degree $2$
Conductor $475$
Sign $0.569 - 0.822i$
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  + (−2.15 − 0.578i)2-s + (−0.677 + 2.52i)3-s + (2.59 + 1.5i)4-s + (2.92 − 5.06i)6-s + (1.22 + 1.22i)7-s + (−1.58 − 1.58i)8-s + (−3.33 − 1.92i)9-s + 4.85·11-s + (−5.55 + 5.55i)12-s + (4.68 − 1.25i)13-s + (−1.93 − 3.35i)14-s + (−0.500 − 0.866i)16-s + (1.17 − 4.38i)17-s + (6.09 + 6.09i)18-s + (4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (−1.52 − 0.409i)2-s + (−0.391 + 1.46i)3-s + (1.29 + 0.750i)4-s + (1.19 − 2.06i)6-s + (0.462 + 0.462i)7-s + (−0.559 − 0.559i)8-s + (−1.11 − 0.642i)9-s + 1.46·11-s + (−1.60 + 1.60i)12-s + (1.30 − 0.348i)13-s + (−0.517 − 0.896i)14-s + (−0.125 − 0.216i)16-s + (0.284 − 1.06i)17-s + (1.43 + 1.43i)18-s + (0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640996 + 0.335748i\)
\(L(\frac12)\) \(\approx\) \(0.640996 + 0.335748i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.33 - 0.5i)T \)
good2 \( 1 + (2.15 + 0.578i)T + (1.73 + i)T^{2} \)
3 \( 1 + (0.677 - 2.52i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 - 4.85T + 11T^{2} \)
13 \( 1 + (-4.68 + 1.25i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.17 + 4.38i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (1.34 + 5.01i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.26 + 3.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (-2.62 + 2.62i)T - 37iT^{2} \)
41 \( 1 + (-0.572 + 0.330i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.12 - 2.17i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.74 - 1.00i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.37 - 1.17i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.25 + 5.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.28 - 9.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.55 - 5.79i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.35 - 1.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.244 + 0.0654i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.992 - 1.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.81 + 7.81i)T - 83iT^{2} \)
89 \( 1 + (6.47 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.1 + 2.99i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.91277415623938315668237524546, −10.17911790828701845187137630438, −9.311337328091976079401095336745, −8.932907369220318330388845830530, −7.992960223808539236856529202211, −6.63808302499979691052947917634, −5.42789015466858328229610003806, −4.28279412657038077344702390824, −3.04476800215168872987705667878, −1.17477598950885987912058227078, 1.12111598249255504373975236066, 1.59782972851092641143125665587, 3.92271819156818529247805662395, 6.01406372389669070910508721573, 6.45046540710209877597096806185, 7.47199415941093048948167036771, 7.944554191537201695876744813860, 8.912108161606176081035793382329, 9.717245711060752000929431497120, 11.04751776646780833465764541860

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