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Properties

Label 2-936-104.101-c1-0-41
Degree $2$
Conductor $936$
Sign $0.620 + 0.783i$
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  + (−1.15 − 0.818i)2-s + (0.659 + 1.88i)4-s + 2.34·5-s + (3.69 + 2.13i)7-s + (0.784 − 2.71i)8-s + (−2.70 − 1.91i)10-s + (−1.33 − 2.31i)11-s + (−2.96 − 2.05i)13-s + (−2.51 − 5.48i)14-s + (−3.12 + 2.49i)16-s + (2.86 − 4.95i)17-s + (3.67 − 6.35i)19-s + (1.54 + 4.42i)20-s + (−0.353 + 3.75i)22-s + (−1.42 − 2.46i)23-s + ⋯
L(s)  = 1  + (−0.815 − 0.578i)2-s + (0.329 + 0.943i)4-s + 1.04·5-s + (1.39 + 0.806i)7-s + (0.277 − 0.960i)8-s + (−0.855 − 0.606i)10-s + (−0.402 − 0.697i)11-s + (−0.821 − 0.570i)13-s + (−0.672 − 1.46i)14-s + (−0.782 + 0.623i)16-s + (0.694 − 1.20i)17-s + (0.842 − 1.45i)19-s + (0.346 + 0.989i)20-s + (−0.0752 + 0.801i)22-s + (−0.296 − 0.514i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.620 + 0.783i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.620 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31961 - 0.638279i\)
\(L(\frac12)\) \(\approx\) \(1.31961 - 0.638279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.818i)T \)
3 \( 1 \)
13 \( 1 + (2.96 + 2.05i)T \)
good5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + (-3.69 - 2.13i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.86 + 4.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.67 + 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.42 + 2.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.01 + 2.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.91iT - 31T^{2} \)
37 \( 1 + (-0.806 - 1.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.52 - 3.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.98 - 2.29i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.83iT - 47T^{2} \)
53 \( 1 + 2.67iT - 53T^{2} \)
59 \( 1 + (-3.87 + 6.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.83 + 1.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.32 - 7.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.6 - 6.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 + (15.0 - 8.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.06 - 2.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.841377365332131565669701396808, −9.247747614346572525478864813117, −8.354056162544480061629230384331, −7.74770394509441098494389075127, −6.68467594214425629197630358752, −5.36075230722845776132181587527, −4.89010560371137725176796670536, −2.92362559112110092220156329849, −2.37748624579232213439399312300, −1.02071153278522434655643085099, 1.45447880747869572309829988281, 2.06195298244950865722296913499, 4.14870362658633280299555646193, 5.25599602307381577135192059531, 5.81269548330057922486612521665, 7.02831584098339947659568004761, 7.74221080520891764969262862979, 8.285146456065928170850051901142, 9.536021549329543842716901231780, 10.07218460772255039435339020069

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