↑

Properties

Label 2-475-19.16-c1-0-13
Degree $2$
Conductor $475$
Sign $0.580 + 0.814i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 1.14i)2-s + (2.24 − 0.815i)3-s + (0.205 + 1.16i)4-s + (−3.99 − 1.45i)6-s + (2.11 + 3.67i)7-s + (−0.727 + 1.25i)8-s + (2.05 − 1.72i)9-s + (0.245 − 0.425i)11-s + (1.41 + 2.44i)12-s + (3.91 + 1.42i)13-s + (1.31 − 7.45i)14-s + (4.66 − 1.69i)16-s + (1.55 + 1.30i)17-s − 4.78·18-s + (1.86 − 3.93i)19-s + ⋯
L(s)  = 1  + (−0.966 − 0.811i)2-s + (1.29 − 0.470i)3-s + (0.102 + 0.583i)4-s + (−1.63 − 0.594i)6-s + (0.801 + 1.38i)7-s + (−0.257 + 0.445i)8-s + (0.684 − 0.574i)9-s + (0.0740 − 0.128i)11-s + (0.407 + 0.706i)12-s + (1.08 + 0.394i)13-s + (0.351 − 1.99i)14-s + (1.16 − 0.424i)16-s + (0.378 + 0.317i)17-s − 1.12·18-s + (0.428 − 0.903i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27585 - 0.657244i\)
\(L(\frac12)\) \(\approx\) \(1.27585 - 0.657244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 3.93i)T \)
good2 \( 1 + (1.36 + 1.14i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (-2.24 + 0.815i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.11 - 3.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.245 + 0.425i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.91 - 1.42i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.55 - 1.30i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.763 + 4.32i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.49 - 2.09i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.04 + 3.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.14T + 37T^{2} \)
41 \( 1 + (4.10 - 1.49i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.84 - 10.4i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.00 - 1.68i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.98 + 11.2i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.415 + 0.348i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.36 + 13.4i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.48 + 4.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.04 + 5.94i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.93 + 0.702i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.01 - 1.82i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.05 - 7.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.07 - 1.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.32 + 3.62i)T + (16.8 + 95.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.02549941374670435463654323458, −9.578857316986071644761276738251, −9.077917799717148762989842515891, −8.315537770012489095681132468590, −7.975272215189326960864205935420, −6.36434550463895392502984466710, −5.13320859364088730961349202443, −3.34168299613396813749135348247, −2.35583367495741038300505094848, −1.54051289457324970681760077081, 1.33445921549206239180217539531, 3.42837808712628525444709773359, 4.02994151050743097564861415882, 5.67617211868899708413289028232, 7.12428301879753418327002616601, 7.73399965410787736430584914682, 8.339825136696688055239564919267, 9.124385835072987150221995338337, 10.01918384839756170193886389524, 10.60220834646664302268473731819

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