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Properties

Label 2-936-104.99-c1-0-55
Degree $2$
Conductor $936$
Sign $0.350 + 0.936i$
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  + (1.18 − 0.776i)2-s + (0.793 − 1.83i)4-s + (2.44 − 2.44i)5-s + (2.97 + 2.97i)7-s + (−0.487 − 2.78i)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 + (−2.55 − 6.43i)20-s + (−1.04 + 5.04i)22-s − 4.08·23-s + ⋯
L(s)  = 1  + (0.835 − 0.549i)2-s + (0.396 − 0.917i)4-s + (1.09 − 1.09i)5-s + (1.12 + 1.12i)7-s + (−0.172 − 0.985i)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 + (−0.570 − 1.43i)20-s + (−0.222 + 1.07i)22-s − 0.850·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.350 + 0.936i$
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: \((2,\ 936,\ (\ :1/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74733 - 1.90431i\)
\(L(\frac12)\) \(\approx\) \(2.74733 - 1.90431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.776i)T \)
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

−10.09870392316433492250570000139, −9.057761604276842372618168875507, −8.533608780711527830893004049425, −7.28545944896970797186268804101, −5.86629269239021522960199786615, −5.20267767667112149064591415277, −5.08593215760832434154090494615, −3.50136022844880671472643195306, −2.03990287148853972674350235815, −1.55405899576756074655249098973, 1.80669807004432757511133171704, 2.98580584035428949038589700149, 4.00468736135630351224746251098, 5.05982178452072209667979702091, 6.01815007406031624351150082251, 6.57232367841951582442728072887, 7.67785216573330708729478476670, 8.075220761219842020782352118733, 9.446050174945601678813436863184, 10.46345151907645170188074067286

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