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Properties

Label 2-936-104.101-c1-0-60
Degree $2$
Conductor $936$
Sign $0.00641 + 0.999i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s − 0.267·5-s + (−3 − 1.73i)7-s + (1.99 − 2i)8-s + (−0.366 + 0.0980i)10-s + (−1 − 1.73i)11-s + (2.59 + 2.5i)13-s + (−4.73 − 1.26i)14-s + (1.99 − 3.46i)16-s + (3.23 − 5.59i)17-s + (2.36 − 4.09i)19-s + (−0.464 + 0.267i)20-s + (−2 − 1.99i)22-s + (−1.09 − 1.90i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s − 0.119·5-s + (−1.13 − 0.654i)7-s + (0.707 − 0.707i)8-s + (−0.115 + 0.0310i)10-s + (−0.301 − 0.522i)11-s + (0.720 + 0.693i)13-s + (−1.26 − 0.338i)14-s + (0.499 − 0.866i)16-s + (0.783 − 1.35i)17-s + (0.542 − 0.940i)19-s + (−0.103 + 0.0599i)20-s + (−0.426 − 0.426i)22-s + (−0.228 − 0.396i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.00641 + 0.999i$
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: \((2,\ 936,\ (\ :1/2),\ 0.00641 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75456 - 1.74335i\)
\(L(\frac12)\) \(\approx\) \(1.75456 - 1.74335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
13 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 + 0.267T + 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (-3.86 - 6.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.03 + 0.598i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.19 + 4.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.26iT - 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 + (3.73 - 6.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 - 6.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + (-0.464 + 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.927691409183327169359277012311, −9.344406681691370826421050342422, −7.949456959349357993508878631163, −7.00396500620774179454701122414, −6.39460065824097365589405066876, −5.44258615311702612758194650688, −4.37780148477006967609613668145, −3.45630077140752615472690908528, −2.67890572817001675659794805249, −0.861396753172196908325327668250, 1.89887000555132024580175752835, 3.29602147932287314197795783424, 3.74192087355765304226224865282, 5.21106432271597557731762028173, 5.94327190136883027051425537831, 6.51046030466102754969834702749, 7.78126065203882477100397371290, 8.236869317565539712142889930415, 9.612575597763893626074719480585, 10.27187932584092401935472616235

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