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Properties

Label 2-936-117.16-c1-0-23
Degree $2$
Conductor $936$
Sign $0.782 + 0.622i$
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  + 1.73i·3-s + (−0.5 − 0.866i)5-s + (−1.5 − 2.59i)7-s − 2.99·9-s + (1 + 3.46i)13-s + (1.49 − 0.866i)15-s + (2.5 − 4.33i)17-s + (0.5 − 0.866i)19-s + (4.5 − 2.59i)21-s + (1.5 − 2.59i)23-s + (2 − 3.46i)25-s − 5.19i·27-s + 6·29-s + (−1.5 − 2.59i)31-s + (−1.5 + 2.59i)35-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.223 − 0.387i)5-s + (−0.566 − 0.981i)7-s − 0.999·9-s + (0.277 + 0.960i)13-s + (0.387 − 0.223i)15-s + (0.606 − 1.05i)17-s + (0.114 − 0.198i)19-s + (0.981 − 0.566i)21-s + (0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s − 0.999i·27-s + 1.11·29-s + (−0.269 − 0.466i)31-s + (−0.253 + 0.439i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13517 - 0.396760i\)
\(L(\frac12)\) \(\approx\) \(1.13517 - 0.396760i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
13 \( 1 + (-1 - 3.46i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.5 + 4.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 + 2.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09142088107626843581816671306, −9.141204460799194671402938269585, −8.593984372617298149350194884767, −7.37648310717874551860172041437, −6.59847807030327268763754114707, −5.42262832671265224186776003544, −4.46654299205487102683747916182, −3.84896098768539731712708208647, −2.71445092466769826791219581240, −0.62269264981030814992835933423, 1.32987840599300460738259607724, 2.78537785686881180000461703212, 3.40225538659686293872379768811, 5.19930387477721635861113527683, 6.00470777901217092375381527203, 6.63519825851573582814029992517, 7.72513777934955731659877762045, 8.295849909400138371500916965594, 9.175018599997553477240256843508, 10.18922484441026859178866815156

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