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Properties

Label 2-936-13.9-c1-0-12
Degree $2$
Conductor $936$
Sign $-0.575 + 0.818i$
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  − 3.17·5-s + (1.08 + 1.88i)7-s + (1.45 − 2.52i)11-s + (−1.21 + 3.39i)13-s + (−3.04 − 5.27i)17-s + (−1.45 − 2.52i)19-s + (0.281 − 0.488i)23-s + 5.09·25-s + (1.32 − 2.30i)29-s − 7.09·31-s + (−3.45 − 5.99i)35-s + (3.76 − 6.52i)37-s + (−1.30 + 2.26i)41-s + (−5.00 − 8.67i)43-s − 3.43·47-s + ⋯
L(s)  = 1  − 1.42·5-s + (0.411 + 0.712i)7-s + (0.439 − 0.762i)11-s + (−0.337 + 0.941i)13-s + (−0.739 − 1.28i)17-s + (−0.334 − 0.579i)19-s + (0.0587 − 0.101i)23-s + 1.01·25-s + (0.246 − 0.427i)29-s − 1.27·31-s + (−0.584 − 1.01i)35-s + (0.619 − 1.07i)37-s + (−0.204 + 0.353i)41-s + (−0.763 − 1.32i)43-s − 0.501·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.575 + 0.818i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.575 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228471 - 0.439953i\)
\(L(\frac12)\) \(\approx\) \(0.228471 - 0.439953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 3.39i)T \)
good5 \( 1 + 3.17T + 5T^{2} \)
7 \( 1 + (-1.08 - 1.88i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.45 + 2.52i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.04 + 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.45 + 2.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.281 + 0.488i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.32 + 2.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.09T + 31T^{2} \)
37 \( 1 + (-3.76 + 6.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.30 - 2.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.00 + 8.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 + (7.27 + 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.39 + 7.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.52 - 9.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.71 - 6.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 9.96T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.629 + 1.09i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.488875644045103387490732202307, −8.891803169913963078626245647077, −8.161894569852843207640565549656, −7.24571930903093917921320494806, −6.52477706257969478525316259903, −5.20085646178555306171078709310, −4.39833149707529973828913761754, −3.44803597520639179846382263660, −2.19533440453654932568018702070, −0.23621612107437885500248268702, 1.52509614412317276971244467712, 3.25816119673642692956014146968, 4.15641709084642720352525109441, 4.75860797723116956804463794867, 6.18179989769101395374354511135, 7.18358582965972441486639306058, 7.85674745713607124943973021352, 8.390424871007786359746725847815, 9.546323828273021323855284000855, 10.62013713752113578352264560418

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