Properties

Label 2-475-475.111-c1-0-3
Degree $2$
Conductor $475$
Sign $-0.608 - 0.793i$
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  + (−1.50 − 0.105i)2-s + (0.0527 + 1.50i)3-s + (0.271 + 0.0381i)4-s + (−2.06 − 0.865i)5-s + (0.0794 − 2.27i)6-s + (1.96 + 3.39i)7-s + (2.54 + 0.541i)8-s + (0.716 − 0.0501i)9-s + (3.01 + 1.51i)10-s + (−0.272 − 2.59i)11-s + (−0.0432 + 0.411i)12-s + (1.35 + 2.01i)13-s + (−2.59 − 5.31i)14-s + (1.19 − 3.15i)15-s + (−4.29 − 1.23i)16-s + (0.613 − 1.51i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.0743i)2-s + (0.0304 + 0.871i)3-s + (0.135 + 0.0190i)4-s + (−0.921 − 0.387i)5-s + (0.0324 − 0.929i)6-s + (0.741 + 1.28i)7-s + (0.900 + 0.191i)8-s + (0.238 − 0.0167i)9-s + (0.951 + 0.480i)10-s + (−0.0822 − 0.782i)11-s + (−0.0124 + 0.118i)12-s + (0.377 + 0.559i)13-s + (−0.693 − 1.42i)14-s + (0.309 − 0.815i)15-s + (−1.07 − 0.308i)16-s + (0.148 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248527 + 0.504067i\)
\(L(\frac12)\) \(\approx\) \(0.248527 + 0.504067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.06 + 0.865i)T \)
19 \( 1 + (2.85 - 3.28i)T \)
good2 \( 1 + (1.50 + 0.105i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (-0.0527 - 1.50i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.96 - 3.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.272 + 2.59i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-1.35 - 2.01i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-0.613 + 1.51i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (1.15 + 1.11i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-3.38 - 8.36i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (3.66 - 4.06i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (9.77 + 7.10i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.15 + 2.05i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (0.970 - 5.50i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.96 - 4.86i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (-2.76 - 0.388i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-1.11 - 4.47i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (6.07 + 5.86i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-9.15 - 5.71i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (3.95 + 2.10i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-0.351 + 0.520i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.100 + 2.86i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (5.60 - 6.22i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-11.7 + 3.36i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-3.15 + 1.97i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.97507666027767585112346943658, −10.48842226817165033373873391272, −9.214659268166116015528614087826, −8.738436438272107157691816132087, −8.248934190935290459155053918871, −7.07010320948941601478377259690, −5.39824544705126716034834185279, −4.64901401281868867819412257639, −3.53230193703621259533350652411, −1.59941124384506522943099922832, 0.53341510529168395150479864334, 1.82763843265407310107806272891, 3.89635380897029093240641077418, 4.68556061292087009557848769869, 6.73492382757028760283899667040, 7.27307822636756995085424002859, 7.962076857275747966477931868218, 8.471424517261385787290314764342, 10.04015547576988600569827369587, 10.46017800332497552665774822680

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