Properties

Label 2-475-475.113-c1-0-26
Degree $2$
Conductor $475$
Sign $0.0854 + 0.996i$
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  + (0.123 + 0.777i)2-s + (−2.41 − 1.22i)3-s + (1.31 − 0.426i)4-s + (−0.0275 − 2.23i)5-s + (0.658 − 2.02i)6-s + (−0.638 + 0.638i)7-s + (1.20 + 2.37i)8-s + (2.54 + 3.50i)9-s + (1.73 − 0.296i)10-s + (3.36 + 2.44i)11-s + (−3.68 − 0.584i)12-s + (0.663 − 4.18i)13-s + (−0.575 − 0.417i)14-s + (−2.68 + 5.42i)15-s + (0.537 − 0.390i)16-s + (−2.72 − 5.34i)17-s + ⋯
L(s)  = 1  + (0.0871 + 0.549i)2-s + (−1.39 − 0.709i)3-s + (0.656 − 0.213i)4-s + (−0.0123 − 0.999i)5-s + (0.268 − 0.827i)6-s + (−0.241 + 0.241i)7-s + (0.427 + 0.838i)8-s + (0.847 + 1.16i)9-s + (0.548 − 0.0938i)10-s + (1.01 + 0.736i)11-s + (−1.06 − 0.168i)12-s + (0.184 − 1.16i)13-s + (−0.153 − 0.111i)14-s + (−0.692 + 1.40i)15-s + (0.134 − 0.0975i)16-s + (−0.660 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0854 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0854 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0854 + 0.996i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0854 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734081 - 0.673824i\)
\(L(\frac12)\) \(\approx\) \(0.734081 - 0.673824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.0275 + 2.23i)T \)
19 \( 1 + (3.17 + 2.98i)T \)
good2 \( 1 + (-0.123 - 0.777i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (2.41 + 1.22i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (0.638 - 0.638i)T - 7iT^{2} \)
11 \( 1 + (-3.36 - 2.44i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.663 + 4.18i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (2.72 + 5.34i)T + (-9.99 + 13.7i)T^{2} \)
23 \( 1 + (0.442 + 2.79i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (2.43 + 7.49i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.561 + 0.182i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.70 + 0.586i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-2.88 - 3.97i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (5.37 + 5.37i)T + 43iT^{2} \)
47 \( 1 + (-8.99 - 4.58i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.710 - 0.362i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-10.0 + 7.31i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.98 + 2.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.68 + 2.38i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (-0.111 + 0.0363i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.12 - 7.09i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.53 - 4.71i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.10 + 1.07i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (14.7 + 10.7i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.00 - 1.97i)T + (-57.0 - 78.4i)T^{2} \)
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

−11.15476709546851052217880294321, −10.02276299924350004267346742712, −8.899989337017401190012502642114, −7.71550755307951920785168529137, −6.87096584115095682641696813948, −6.16702506261983338552481656450, −5.36825479713290582875336464708, −4.52892196669979892450832823625, −2.17856077207713866950692848383, −0.70408923160363398280004236598, 1.76502335723699193344599920978, 3.64388394116692327771412489862, 4.05667383286560331818215598869, 5.83607364731333186142217227371, 6.49817157549601379907351054414, 7.06156677046525144688578597702, 8.744843259110705774650181660847, 9.997020624318292893285451329614, 10.63811836604210765410940043634, 11.16299998138970058871785409359

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