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Properties

Label 2-936-24.11-c1-0-5
Degree $2$
Conductor $936$
Sign $-0.973 + 0.228i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.01 + 0.979i)2-s + (0.0801 + 1.99i)4-s − 2.22·5-s − 0.534i·7-s + (−1.87 + 2.11i)8-s + (−2.26 − 2.17i)10-s + 2.35i·11-s − i·13-s + (0.523 − 0.545i)14-s + (−3.98 + 0.320i)16-s + 4.78i·17-s − 4.25·19-s + (−0.178 − 4.44i)20-s + (−2.30 + 2.40i)22-s − 2.55·23-s + ⋯
L(s)  = 1  + (0.721 + 0.692i)2-s + (0.0400 + 0.999i)4-s − 0.993·5-s − 0.202i·7-s + (−0.663 + 0.748i)8-s + (−0.716 − 0.688i)10-s + 0.710i·11-s − 0.277i·13-s + (0.139 − 0.145i)14-s + (−0.996 + 0.0801i)16-s + 1.16i·17-s − 0.975·19-s + (−0.0398 − 0.993i)20-s + (−0.491 + 0.512i)22-s − 0.532·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106173 - 0.918906i\)
\(L(\frac12)\) \(\approx\) \(0.106173 - 0.918906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 - 0.979i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 2.22T + 5T^{2} \)
7 \( 1 + 0.534iT - 7T^{2} \)
11 \( 1 - 2.35iT - 11T^{2} \)
17 \( 1 - 4.78iT - 17T^{2} \)
19 \( 1 + 4.25T + 19T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 + 8.32T + 29T^{2} \)
31 \( 1 - 0.284iT - 31T^{2} \)
37 \( 1 + 6.30iT - 37T^{2} \)
41 \( 1 - 4.33iT - 41T^{2} \)
43 \( 1 + 0.666T + 43T^{2} \)
47 \( 1 + 1.10T + 47T^{2} \)
53 \( 1 + 1.42T + 53T^{2} \)
59 \( 1 - 3.21iT - 59T^{2} \)
61 \( 1 - 4.98iT - 61T^{2} \)
67 \( 1 - 0.658T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 - 1.91iT - 89T^{2} \)
97 \( 1 + 7.69T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73716319002032762714347191820, −9.545621165394081193128878860512, −8.445631430010926909422512971469, −7.87097092930241516639709649120, −7.15086198424856075333345884999, −6.23833696140220692435109853168, −5.28295053191865045181448290999, −4.06462642319328186568583648009, −3.81993803806590009471248812511, −2.22390249345124140780934908337, 0.32237907170599647530537156028, 2.08494182571690313621215324199, 3.30706791961545724088796984890, 4.05713987563500915329869248028, 5.01096195685742815795482048251, 5.98073811322629215438840090895, 6.93422557660716592186342047235, 7.947009647575314428161449459861, 8.917237061961999720188152231510, 9.709269831000803227449280905471

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