Properties

Label 2-936-104.51-c0-0-5
Degree $2$
Conductor $936$
Sign $-0.707 + 0.707i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s − 1.41·5-s + (−0.707 − 0.707i)8-s + (−1.00 + 1.00i)10-s − 1.41i·11-s i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 − 1.00i)22-s + 1.00·25-s + (−0.707 − 0.707i)26-s + (−0.707 + 0.707i)32-s + (1.00 + 1.00i)40-s + 1.41i·41-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s − 1.41·5-s + (−0.707 − 0.707i)8-s + (−1.00 + 1.00i)10-s − 1.41i·11-s i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 − 1.00i)22-s + 1.00·25-s + (−0.707 − 0.707i)26-s + (−0.707 + 0.707i)32-s + (1.00 + 1.00i)40-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004161556\)
\(L(\frac12)\) \(\approx\) \(1.004161556\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.30057413330274956547939167049, −9.140141055400086623878744426553, −8.265147077845028517519185684863, −7.51010560449237694889688679136, −6.27718945984503453067121096975, −5.46232096197144585343636146710, −4.37230122847585154368583145156, −3.53762661819863578054090412617, −2.81303690122289617830950363093, −0.799158611783977020561209261077, 2.35723789587639709201561126128, 3.82431174700028423177915923583, 4.27447883147947906189345472963, 5.18071417885716455389129283630, 6.47752045783961125306666720327, 7.28657436058804258769550037318, 7.68770928382644583689887366905, 8.712802123389830418130725114453, 9.499626245371266361180892271581, 10.84399028364160447427385692993

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