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Properties

Label 2-936-104.99-c1-0-67
Degree $2$
Conductor $936$
Sign $-0.999 + 0.0122i$
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  + (0.776 − 1.18i)2-s + (−0.793 − 1.83i)4-s + (2.44 − 2.44i)5-s + (−2.97 − 2.97i)7-s + (−2.78 − 0.487i)8-s + (−0.992 − 4.79i)10-s + (2.57 − 2.57i)11-s + (−3.19 + 1.66i)13-s + (−5.82 + 1.20i)14-s + (−2.73 + 2.91i)16-s + 1.68i·17-s + (3.97 + 3.97i)19-s + (−6.43 − 2.55i)20-s + (−1.04 − 5.04i)22-s − 4.08·23-s + ⋯
L(s)  = 1  + (0.549 − 0.835i)2-s + (−0.396 − 0.917i)4-s + (1.09 − 1.09i)5-s + (−1.12 − 1.12i)7-s + (−0.985 − 0.172i)8-s + (−0.313 − 1.51i)10-s + (0.777 − 0.777i)11-s + (−0.887 + 0.461i)13-s + (−1.55 + 0.322i)14-s + (−0.684 + 0.728i)16-s + 0.408i·17-s + (0.911 + 0.911i)19-s + (−1.43 − 0.570i)20-s + (−0.222 − 1.07i)22-s − 0.850·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.999 + 0.0122i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.999 + 0.0122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0115983 - 1.89912i\)
\(L(\frac12)\) \(\approx\) \(0.0115983 - 1.89912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.776 + 1.18i)T \)
3 \( 1 \)
13 \( 1 + (3.19 - 1.66i)T \)
good5 \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \)
7 \( 1 + (2.97 + 2.97i)T + 7iT^{2} \)
11 \( 1 + (-2.57 + 2.57i)T - 11iT^{2} \)
17 \( 1 - 1.68iT - 17T^{2} \)
19 \( 1 + (-3.97 - 3.97i)T + 19iT^{2} \)
23 \( 1 + 4.08T + 23T^{2} \)
29 \( 1 - 7.65iT - 29T^{2} \)
31 \( 1 + (-7.09 + 7.09i)T - 31iT^{2} \)
37 \( 1 + (-3.55 - 3.55i)T + 37iT^{2} \)
41 \( 1 + (-4.64 - 4.64i)T + 41iT^{2} \)
43 \( 1 + 7.70iT - 43T^{2} \)
47 \( 1 + (6.34 + 6.34i)T + 47iT^{2} \)
53 \( 1 + 8.07iT - 53T^{2} \)
59 \( 1 + (-2.25 + 2.25i)T - 59iT^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 + (3.21 + 3.21i)T + 67iT^{2} \)
71 \( 1 + (-2.81 + 2.81i)T - 71iT^{2} \)
73 \( 1 + (-0.407 + 0.407i)T - 73iT^{2} \)
79 \( 1 - 3.13iT - 79T^{2} \)
83 \( 1 + (3.46 + 3.46i)T + 83iT^{2} \)
89 \( 1 + (9.16 - 9.16i)T - 89iT^{2} \)
97 \( 1 + (-0.0406 - 0.0406i)T + 97iT^{2} \)
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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

−9.792634173974108869159338154900, −9.299388211108273415777985101378, −8.193731627612274974530374454511, −6.68087721086212297830955610591, −6.06623195351682887653346630622, −5.13642240703844759561949297238, −4.13450710380371864979500778756, −3.29936000156101210241569103469, −1.82276562780139865255726484599, −0.74080688450359742478448363709, 2.60390141271792459825128153036, 2.87828614270983201458841626865, 4.44066491139315751662226709715, 5.61074418382037596301814034912, 6.17606332349970644788497479032, 6.84950749516974812321351323400, 7.59153648879427828786213923422, 8.974423489343924951525847197146, 9.647730382747830093807765337356, 10.01785052211947636369664414334

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