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Properties

Label 2-475-25.9-c1-0-24
Degree $2$
Conductor $475$
Sign $0.881 - 0.472i$
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  + (−0.262 + 0.361i)2-s + (2.64 − 0.860i)3-s + (0.556 + 1.71i)4-s + (0.0123 − 2.23i)5-s + (−0.384 + 1.18i)6-s + 4.32i·7-s + (−1.61 − 0.525i)8-s + (3.84 − 2.79i)9-s + (0.805 + 0.592i)10-s + (−0.742 − 0.539i)11-s + (2.94 + 4.05i)12-s + (1.76 + 2.43i)13-s + (−1.56 − 1.13i)14-s + (−1.89 − 5.93i)15-s + (−2.29 + 1.66i)16-s + (2.92 + 0.948i)17-s + ⋯
L(s)  = 1  + (−0.185 + 0.255i)2-s + (1.52 − 0.496i)3-s + (0.278 + 0.856i)4-s + (0.00553 − 0.999i)5-s + (−0.157 + 0.483i)6-s + 1.63i·7-s + (−0.571 − 0.185i)8-s + (1.28 − 0.931i)9-s + (0.254 + 0.187i)10-s + (−0.223 − 0.162i)11-s + (0.850 + 1.17i)12-s + (0.490 + 0.675i)13-s + (−0.418 − 0.303i)14-s + (−0.488 − 1.53i)15-s + (−0.574 + 0.417i)16-s + (0.708 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12535 + 0.533182i\)
\(L(\frac12)\) \(\approx\) \(2.12535 + 0.533182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0123 + 2.23i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.262 - 0.361i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.64 + 0.860i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 - 4.32iT - 7T^{2} \)
11 \( 1 + (0.742 + 0.539i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.76 - 2.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.92 - 0.948i)T + (13.7 + 9.99i)T^{2} \)
23 \( 1 + (-3.67 + 5.06i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.90 + 5.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.458 - 1.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.89 + 5.36i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.95 + 1.42i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.63iT - 43T^{2} \)
47 \( 1 + (10.0 - 3.27i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.90 - 1.59i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.68 - 4.12i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-6.29 - 4.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-8.04 - 2.61i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.62 - 11.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.55 + 10.3i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.35 - 10.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.0 + 3.27i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.85 + 2.80i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.493 - 0.160i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.37657343596796962449946349164, −9.530706577320627626568212889882, −8.891403949638187998874042688987, −8.484793319708933669503937725332, −7.85237282368437287115593009776, −6.70537675432843367603608888158, −5.51019892078526009032998233569, −3.97958120971127140482238182459, −2.84164846071382192802203691505, −1.96026313754116632374652129347, 1.51335967138465575178449618317, 3.07912058217488582457784872335, 3.57261171709917307932723989389, 5.04880365482599729906602409811, 6.57323012373801888846337083478, 7.45315136900965549633857022063, 8.153780856084427283859849479420, 9.641987302429453763307437073990, 9.861483100495049013450948757998, 10.75466263290498881349083279711

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