↑

Properties

Label 2-1295-7.4-c1-0-94
Degree $2$
Conductor $1295$
Sign $-0.251 - 0.967i$
Analytic cond. $10.3406$
Root an. cond. $3.21568$
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  + (−0.439 − 0.761i)2-s + (0.702 − 1.21i)3-s + (0.613 − 1.06i)4-s + (−0.5 − 0.866i)5-s − 1.23·6-s + (−1.97 − 1.76i)7-s − 2.83·8-s + (0.514 + 0.890i)9-s + (−0.439 + 0.761i)10-s + (−1.30 + 2.26i)11-s + (−0.860 − 1.49i)12-s − 1.85·13-s + (−0.475 + 2.27i)14-s − 1.40·15-s + (0.0222 + 0.0385i)16-s + (−0.331 + 0.573i)17-s + ⋯
L(s)  = 1  + (−0.311 − 0.538i)2-s + (0.405 − 0.702i)3-s + (0.306 − 0.530i)4-s + (−0.223 − 0.387i)5-s − 0.504·6-s + (−0.745 − 0.666i)7-s − 1.00·8-s + (0.171 + 0.296i)9-s + (−0.139 + 0.240i)10-s + (−0.394 + 0.683i)11-s + (−0.248 − 0.430i)12-s − 0.514·13-s + (−0.127 + 0.608i)14-s − 0.362·15-s + (0.00556 + 0.00964i)16-s + (−0.0803 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $-0.251 - 0.967i$
Analytic conductor: \(10.3406\)
Root analytic conductor: \(3.21568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1295} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1295,\ (\ :1/2),\ -0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4498017391\)
\(L(\frac12)\) \(\approx\) \(0.4498017391\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.97 + 1.76i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.439 + 0.761i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.702 + 1.21i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.30 - 2.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.85T + 13T^{2} \)
17 \( 1 + (0.331 - 0.573i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.727 - 1.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.25 + 3.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + 5.51T + 41T^{2} \)
43 \( 1 - 4.15T + 43T^{2} \)
47 \( 1 + (1.35 + 2.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.93 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.47 - 7.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.25 + 5.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.610 + 1.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (-1.15 + 2.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.61 + 4.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (3.80 + 6.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.968T + 97T^{2} \)
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

−9.250704005171113448873390238494, −8.272846100193563807500973360702, −7.34900152062507390790002000078, −6.89588184563192117420476173995, −5.80116460866843701585812346364, −4.77590541020750578337236042416, −3.58176266914387962246765047891, −2.39927817565684699497512956216, −1.60811393010505494542583842416, −0.18121941819249932900879265674, 2.47035274106600934478190350853, 3.28579908714294688200317538748, 3.91796429329525102580802893198, 5.41682107884457350062528323592, 6.18867462183746677275658468616, 7.09602979810636894710501199495, 7.76183450298044964772951712843, 8.714159672226003854856397558888, 9.322474831557907600421823845651, 9.876722293521982997398156358496

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