↑

Properties

Label 2-475-19.9-c1-0-15
Degree $2$
Conductor $475$
Sign $0.682 + 0.730i$
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  + (−0.0424 − 0.240i)2-s + (−2.09 + 1.76i)3-s + (1.82 − 0.663i)4-s + (0.512 + 0.430i)6-s + (0.970 − 1.68i)7-s + (−0.481 − 0.834i)8-s + (0.781 − 4.43i)9-s + (−2.11 − 3.66i)11-s + (−2.65 + 4.60i)12-s + (−0.972 − 0.816i)13-s + (−0.445 − 0.162i)14-s + (2.79 − 2.34i)16-s + (−0.427 − 2.42i)17-s − 1.09·18-s + (1.64 + 4.03i)19-s + ⋯
L(s)  = 1  + (−0.0300 − 0.170i)2-s + (−1.21 + 1.01i)3-s + (0.911 − 0.331i)4-s + (0.209 + 0.175i)6-s + (0.366 − 0.635i)7-s + (−0.170 − 0.294i)8-s + (0.260 − 1.47i)9-s + (−0.638 − 1.10i)11-s + (−0.766 + 1.32i)12-s + (−0.269 − 0.226i)13-s + (−0.119 − 0.0433i)14-s + (0.698 − 0.585i)16-s + (−0.103 − 0.587i)17-s − 0.259·18-s + (0.376 + 0.926i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.682 + 0.730i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978281 - 0.424945i\)
\(L(\frac12)\) \(\approx\) \(0.978281 - 0.424945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 4.03i)T \)
good2 \( 1 + (0.0424 + 0.240i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (2.09 - 1.76i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.970 + 1.68i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.972 + 0.816i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.427 + 2.42i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.82 + 1.75i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.50 + 8.50i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.55 - 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + (-1.91 + 1.60i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.19 + 1.52i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.17 + 6.66i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.648 - 0.235i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.901 - 5.11i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.63 - 0.958i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.905 - 5.13i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.744 + 0.270i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.08 + 0.910i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.64 - 3.89i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.92 - 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.71 + 1.43i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.49 + 8.47i)T + (-91.1 + 33.1i)T^{2} \)
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

−10.89633643534413095323971482812, −10.32233142057446724975817567765, −9.582377351538674559606059661413, −8.093099649618479367584343363732, −7.04026791400604504989899438332, −5.93130521477799987867622850642, −5.36079405974104719492209123574, −4.21188090634869447972939527762, −2.88846291941789091683162211542, −0.78698337669179878761797455789, 1.59368135210814301254841407140, 2.69020048620959647856641138866, 4.81764479935049383594145270816, 5.65301851113922100025349090360, 6.63257302144277725282193947432, 7.27038817462093324860448265972, 7.977030575985449854589293215914, 9.322931050536428299629311545142, 10.69340017654580270466223307837, 11.28875227072891967196355295866

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