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Properties

Label 2-936-39.32-c1-0-4
Degree $2$
Conductor $936$
Sign $0.177 - 0.984i$
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  + (2.43 + 2.43i)5-s + (−0.0623 + 0.232i)7-s + (0.655 + 2.44i)11-s + (0.765 + 3.52i)13-s + (0.156 + 0.270i)17-s + (−4.72 − 1.26i)19-s + (1.56 − 2.70i)23-s + 6.84i·25-s + (0.251 + 0.145i)29-s + (−2.38 + 2.38i)31-s + (−0.717 + 0.414i)35-s + (−2.72 + 0.730i)37-s + (5.24 − 1.40i)41-s + (−7.61 + 4.39i)43-s + (9.47 − 9.47i)47-s + ⋯
L(s)  = 1  + (1.08 + 1.08i)5-s + (−0.0235 + 0.0878i)7-s + (0.197 + 0.737i)11-s + (0.212 + 0.977i)13-s + (0.0378 + 0.0655i)17-s + (−1.08 − 0.290i)19-s + (0.326 − 0.565i)23-s + 1.36i·25-s + (0.0466 + 0.0269i)29-s + (−0.427 + 0.427i)31-s + (−0.121 + 0.0700i)35-s + (−0.448 + 0.120i)37-s + (0.819 − 0.219i)41-s + (−1.16 + 0.670i)43-s + (1.38 − 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38918 + 1.16119i\)
\(L(\frac12)\) \(\approx\) \(1.38918 + 1.16119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-0.765 - 3.52i)T \)
good5 \( 1 + (-2.43 - 2.43i)T + 5iT^{2} \)
7 \( 1 + (0.0623 - 0.232i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.655 - 2.44i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.156 - 0.270i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.72 + 1.26i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.56 + 2.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.251 - 0.145i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.38 - 2.38i)T - 31iT^{2} \)
37 \( 1 + (2.72 - 0.730i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.24 + 1.40i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.61 - 4.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.47 + 9.47i)T - 47iT^{2} \)
53 \( 1 - 7.02iT - 53T^{2} \)
59 \( 1 + (4.77 + 1.28i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.10 - 1.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0954 + 0.356i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.10 + 7.84i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 73iT^{2} \)
79 \( 1 - 9.82T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + (-3.43 - 12.8i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.7 - 2.86i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.47634659534129371288254751303, −9.364496036092380009292077309196, −8.871786149116413435761674405187, −7.49808050755208648464395667501, −6.64531515087820503252076409179, −6.24188819819334776774013883702, −5.02508196931163738408796156946, −3.93387266707176818505828712689, −2.60722818423463892706339148879, −1.82175507435275589396155163765, 0.870297577756179951669129821198, 2.11433310989282674575220159123, 3.49741776876675517727243917886, 4.69688618392329317999597022590, 5.66690608818552736529395091612, 6.08640910472133363742004158945, 7.40662173334387907515932894452, 8.500104753053459921737894616993, 8.914528186968929046243254537523, 9.861835157637459501183411230748

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