Properties

Label 2-936-104.77-c1-0-12
Degree $2$
Conductor $936$
Sign $-0.980 + 0.196i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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 + i)2-s + 2i·4-s − 5-s + 3i·7-s + (−2 + 2i)8-s + (−1 − i)10-s − 2·11-s + (−3 − 2i)13-s + (−3 + 3i)14-s − 4·16-s − 3·17-s − 2i·20-s + (−2 − 2i)22-s + 6·23-s − 4·25-s + (−1 − 5i)26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s − 0.447·5-s + 1.13i·7-s + (−0.707 + 0.707i)8-s + (−0.316 − 0.316i)10-s − 0.603·11-s + (−0.832 − 0.554i)13-s + (−0.801 + 0.801i)14-s − 16-s − 0.727·17-s − 0.447i·20-s + (−0.426 − 0.426i)22-s + 1.25·23-s − 0.800·25-s + (−0.196 − 0.980i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.980 + 0.196i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.980 + 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116245 - 1.17396i\)
\(L(\frac12)\) \(\approx\) \(0.116245 - 1.17396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
13 \( 1 + (3 + 2i)T \)
good5 \( 1 + T + 5T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 9iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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

−10.65612740236107695790796582103, −9.367058236941436946838956909401, −8.677899163356811975775206407300, −7.83881314973015279409779516984, −7.13176885768600318265574262979, −6.09090916221809287640824532885, −5.25323210754301293279961185377, −4.59174313634459510366590335032, −3.22520343473956141973312925965, −2.43280148417257110668833704133, 0.41005069647730060555325328768, 2.04571798562098403577891944973, 3.23257123115486763657558912311, 4.29353286187484130695075389112, 4.79578636324188286305997361737, 6.05299920579516037395756635088, 7.06301245441252675334246236615, 7.71697366843732888175956990579, 9.051255729462906967283285122444, 9.838685379266236219076443981276

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