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Properties

Label 2-936-104.99-c1-0-35
Degree $2$
Conductor $936$
Sign $0.289 - 0.957i$
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 + i)2-s + 2i·4-s + (2.54 − 2.54i)5-s + (2.54 + 2.54i)7-s + (−2 + 2i)8-s + 5.09·10-s + (−1 + i)11-s + (−2.54 + 2.54i)13-s + 5.09i·14-s − 4·16-s − 3i·17-s + (2 + 2i)19-s + (5.09 + 5.09i)20-s − 2·22-s + 5.09·23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (1.14 − 1.14i)5-s + (0.963 + 0.963i)7-s + (−0.707 + 0.707i)8-s + 1.61·10-s + (−0.301 + 0.301i)11-s + (−0.707 + 0.707i)13-s + 1.36i·14-s − 16-s − 0.727i·17-s + (0.458 + 0.458i)19-s + (1.14 + 1.14i)20-s − 0.426·22-s + 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.289 - 0.957i$
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: \((2,\ 936,\ (\ :1/2),\ 0.289 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33683 + 1.73406i\)
\(L(\frac12)\) \(\approx\) \(2.33683 + 1.73406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
13 \( 1 + (2.54 - 2.54i)T \)
good5 \( 1 + (-2.54 + 2.54i)T - 5iT^{2} \)
7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 - 5.09iT - 29T^{2} \)
31 \( 1 + (-5.09 + 5.09i)T - 31iT^{2} \)
37 \( 1 + (-2.54 - 2.54i)T + 37iT^{2} \)
41 \( 1 + (6 + 6i)T + 41iT^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (2.54 + 2.54i)T + 47iT^{2} \)
53 \( 1 + 5.09iT - 53T^{2} \)
59 \( 1 + (8 - 8i)T - 59iT^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + (-7.64 + 7.64i)T - 71iT^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 + 5.09iT - 79T^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 + (-2 + 2i)T - 89iT^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−9.937949912302899723180063404412, −9.122256044269451672589827430802, −8.619512165212573419724487081709, −7.68209547190769230461329148660, −6.67370875336032050667200054176, −5.55827311868999934229248287371, −5.11849672562400602854232721136, −4.52823294538890658032763869976, −2.74035986022406899856085015498, −1.74750822922777253876146994724, 1.25558492530026760144186621694, 2.48765451709671433120912480656, 3.26888033588431467602705817501, 4.60704097818432399880515948968, 5.37097271656677812342122588842, 6.33738263526808625532998095193, 7.11184405366992576233665452012, 8.142846438640600119715787099806, 9.557353044162453250250627797339, 10.12966773014694335943590476120

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