↑

Properties

Label 2-475-19.7-c1-0-10
Degree $2$
Conductor $475$
Sign $-0.948 + 0.315i$
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  + (−1.22 − 2.12i)2-s + (−0.780 − 1.35i)3-s + (−2.01 + 3.49i)4-s + (−1.91 + 3.32i)6-s + 4.50·7-s + 4.99·8-s + (0.281 − 0.487i)9-s + 2.19·11-s + 6.29·12-s + (1.87 − 3.25i)13-s + (−5.53 − 9.58i)14-s + (−2.09 − 3.63i)16-s + (−0.332 − 0.576i)17-s − 1.38·18-s + (3.79 + 2.13i)19-s + ⋯
L(s)  = 1  + (−0.868 − 1.50i)2-s + (−0.450 − 0.780i)3-s + (−1.00 + 1.74i)4-s + (−0.782 + 1.35i)6-s + 1.70·7-s + 1.76·8-s + (0.0938 − 0.162i)9-s + 0.662·11-s + 1.81·12-s + (0.521 − 0.902i)13-s + (−1.47 − 2.56i)14-s + (−0.524 − 0.909i)16-s + (−0.0807 − 0.139i)17-s − 0.326·18-s + (0.871 + 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.948 + 0.315i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.948 + 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151359 - 0.933597i\)
\(L(\frac12)\) \(\approx\) \(0.151359 - 0.933597i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.79 - 2.13i)T \)
good2 \( 1 + (1.22 + 2.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.780 + 1.35i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.50T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + (-1.87 + 3.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.332 + 0.576i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.244 + 0.422i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.79 - 3.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.83T + 31T^{2} \)
37 \( 1 + 3.01T + 37T^{2} \)
41 \( 1 + (0.0362 + 0.0627i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.210 + 0.364i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.51 - 4.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.30 - 2.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.26 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.53 - 6.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.86 - 4.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.48 + 6.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.47 - 2.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.66 + 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (-0.668 + 1.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.19 + 3.79i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.83603386899090393439496620544, −9.925809683527665921346949974574, −8.867136275793340915554217266105, −8.103538154358138407401596916266, −7.35789003216309429208741365000, −5.88547756114063839115563711920, −4.54359118244875741963984204513, −3.26735051691760096730612472332, −1.70252752471780801722845134413, −1.03881526281933039060097398099, 1.47173443129065245380355518329, 4.28755976434492210579899386278, 4.95791070482832818390299263301, 5.85256553606502519353441758479, 6.95586172433998153515088097974, 7.82432326392406435797571094324, 8.608080419611848509515301797825, 9.384894082490634637930547433905, 10.27493059575800871664701003152, 11.24470638426130826436757725980

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