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Properties

Label 2-475-95.9-c1-0-25
Degree $2$
Conductor $475$
Sign $0.341 + 0.939i$
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  + (2.58 − 0.455i)2-s + (−0.597 − 0.711i)3-s + (4.58 − 1.66i)4-s + (−1.86 − 1.56i)6-s + (−3.31 − 1.91i)7-s + (6.53 − 3.77i)8-s + (0.371 − 2.10i)9-s + (1.63 + 2.82i)11-s + (−3.92 − 2.26i)12-s + (0.234 − 0.278i)13-s + (−9.44 − 3.43i)14-s + (7.67 − 6.44i)16-s + (−0.673 + 0.118i)17-s − 5.60i·18-s + (3.86 + 2.01i)19-s + ⋯
L(s)  = 1  + (1.82 − 0.321i)2-s + (−0.344 − 0.410i)3-s + (2.29 − 0.833i)4-s + (−0.761 − 0.639i)6-s + (−1.25 − 0.724i)7-s + (2.30 − 1.33i)8-s + (0.123 − 0.701i)9-s + (0.492 + 0.853i)11-s + (−1.13 − 0.653i)12-s + (0.0649 − 0.0773i)13-s + (−2.52 − 0.918i)14-s + (1.91 − 1.61i)16-s + (−0.163 + 0.0288i)17-s − 1.32i·18-s + (0.886 + 0.461i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.341 + 0.939i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.341 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77216 - 1.94241i\)
\(L(\frac12)\) \(\approx\) \(2.77216 - 1.94241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.86 - 2.01i)T \)
good2 \( 1 + (-2.58 + 0.455i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.597 + 0.711i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (3.31 + 1.91i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.63 - 2.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.234 + 0.278i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.673 - 0.118i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-3.20 - 8.80i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.267 - 1.51i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.22 + 2.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.163iT - 37T^{2} \)
41 \( 1 + (5.64 - 4.73i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.885 + 2.43i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (8.66 + 1.52i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.80 + 10.4i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.35 - 7.67i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.27 - 0.827i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-9.41 - 1.65i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.81 - 1.75i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.06 + 6.03i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-7.14 + 5.99i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.631 + 0.364i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.68 + 8.12i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (7.43 - 1.31i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.39759632397158543728465977905, −10.09133213138965391329790755776, −9.521002349709157074810424559650, −7.37114146613426814029338565309, −6.81767768020881153729527246069, −6.09430953116259447690943857675, −5.06228292680674714044247082061, −3.77963258781365940106964360764, −3.27933623263074713969021179387, −1.48465478429656391854419743031, 2.58368945773848486611364172962, 3.42211990345161623604490137846, 4.59670051193064701828409452563, 5.42544017742044913922165185783, 6.29316842968618124972426267256, 6.89454700408789703877304055094, 8.298121539032955488688063383774, 9.516988340588945016766392316225, 10.72634493540076102549548090825, 11.40277066692953322954693939621

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