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Properties

Label 2-936-24.11-c1-0-33
Degree $2$
Conductor $936$
Sign $0.770 - 0.636i$
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 + 0.812i)2-s + (0.680 + 1.88i)4-s + 3.23·5-s − 4.33i·7-s + (−0.740 + 2.72i)8-s + (3.74 + 2.63i)10-s + 1.18i·11-s + i·13-s + (3.52 − 5.01i)14-s + (−3.07 + 2.55i)16-s + 0.0944i·17-s + 2.27·19-s + (2.20 + 6.08i)20-s + (−0.964 + 1.37i)22-s + 7.64·23-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s + (0.340 + 0.940i)4-s + 1.44·5-s − 1.63i·7-s + (−0.261 + 0.965i)8-s + (1.18 + 0.831i)10-s + 0.357i·11-s + 0.277i·13-s + (0.941 − 1.34i)14-s + (−0.768 + 0.639i)16-s + 0.0229i·17-s + 0.522·19-s + (0.492 + 1.36i)20-s + (−0.205 + 0.293i)22-s + 1.59·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.770 - 0.636i$
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: \((2,\ 936,\ (\ :1/2),\ 0.770 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.02313 + 1.08739i\)
\(L(\frac12)\) \(\approx\) \(3.02313 + 1.08739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.812i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 4.33iT - 7T^{2} \)
11 \( 1 - 1.18iT - 11T^{2} \)
17 \( 1 - 0.0944iT - 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
23 \( 1 - 7.64T + 23T^{2} \)
29 \( 1 + 4.80T + 29T^{2} \)
31 \( 1 + 7.57iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 3.47iT - 41T^{2} \)
43 \( 1 + 9.77T + 43T^{2} \)
47 \( 1 + 1.62T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 5.55iT - 59T^{2} \)
61 \( 1 - 8.82iT - 61T^{2} \)
67 \( 1 - 0.890T + 67T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + 6.08iT - 83T^{2} \)
89 \( 1 - 2.52iT - 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−10.09493482604654319256974134503, −9.445619634313664306576142609073, −8.306655493337440271907124250743, −7.15723377599716207280188192754, −6.85493134157527746026566215605, −5.78312452603069307344386642405, −4.95235372823759069643410294517, −4.04846925805491635378313566992, −2.90241228809172212133042890780, −1.52605856208847477862760747193, 1.54804298045178591064348966127, 2.50605146911857183269759466728, 3.27662512702410584493897774934, 5.02839473476696773854926206458, 5.49660701205750622842535713408, 6.09728116151117573255515981516, 7.08489801063922487601677496416, 8.780053584663579824855770121368, 9.201875795024369539423040243672, 10.01581126534278122684653350080

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