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Properties

Label 2-936-936.259-c0-0-5
Degree $2$
Conductor $936$
Sign $-0.939 - 0.342i$
Analytic cond. $0.467124$
Root an. cond. $0.683465$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.499 − 0.866i)4-s + (−0.939 − 1.62i)5-s + (−0.766 + 0.642i)6-s + (0.766 − 1.32i)7-s − 0.999·8-s + (0.766 + 0.642i)9-s − 1.87·10-s + (0.173 + 0.984i)12-s + (0.5 + 0.866i)13-s + (−0.766 − 1.32i)14-s + (0.326 + 1.85i)15-s + (−0.5 + 0.866i)16-s + 0.347·17-s + (0.939 − 0.342i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.499 − 0.866i)4-s + (−0.939 − 1.62i)5-s + (−0.766 + 0.642i)6-s + (0.766 − 1.32i)7-s − 0.999·8-s + (0.766 + 0.642i)9-s − 1.87·10-s + (0.173 + 0.984i)12-s + (0.5 + 0.866i)13-s + (−0.766 − 1.32i)14-s + (0.326 + 1.85i)15-s + (−0.5 + 0.866i)16-s + 0.347·17-s + (0.939 − 0.342i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7829343604\)
\(L(\frac12)\) \(\approx\) \(0.7829343604\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.87T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.02782675169778156711596192984, −9.093351848925111507215819413577, −8.074066293817962184505618455251, −7.36124775210213413333693858939, −6.05556395039115037978779643388, −5.04695070286142868900705522910, −4.35977116882146584676650800872, −3.95458874986993422995310947962, −1.60847339490165730302078584053, −0.808987503715386013813398570999, 2.78134904341040723476938841467, 3.68272497921005082119791578492, 4.74663196531483022735762890125, 5.74506797436249392240218799845, 6.24658725340448499263793813613, 7.26613880254717461187080447691, 7.913120655193262072471962135528, 8.820180092726780767349870938092, 10.04387495777905148489653397552, 10.99255384678648843832044051747

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