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Properties

Label 2-936-24.11-c1-0-15
Degree $2$
Conductor $936$
Sign $0.468 + 0.883i$
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  + (−0.508 − 1.31i)2-s + (−1.48 + 1.34i)4-s − 2.44·5-s − 2.55i·7-s + (2.52 + 1.27i)8-s + (1.24 + 3.22i)10-s + 5.76i·11-s − i·13-s + (−3.36 + 1.29i)14-s + (0.397 − 3.98i)16-s + 1.56i·17-s + 5.82·19-s + (3.62 − 3.28i)20-s + (7.61 − 2.93i)22-s − 2.69·23-s + ⋯
L(s)  = 1  + (−0.359 − 0.933i)2-s + (−0.741 + 0.671i)4-s − 1.09·5-s − 0.964i·7-s + (0.892 + 0.450i)8-s + (0.393 + 1.02i)10-s + 1.73i·11-s − 0.277i·13-s + (−0.900 + 0.346i)14-s + (0.0993 − 0.995i)16-s + 0.379i·17-s + 1.33·19-s + (0.810 − 0.733i)20-s + (1.62 − 0.625i)22-s − 0.561·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794631 - 0.477867i\)
\(L(\frac12)\) \(\approx\) \(0.794631 - 0.477867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.508 + 1.31i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 2.55iT - 7T^{2} \)
11 \( 1 - 5.76iT - 11T^{2} \)
17 \( 1 - 1.56iT - 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 - 6.76T + 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 - 1.74iT - 37T^{2} \)
41 \( 1 + 7.99iT - 41T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 - 3.40T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 9.66iT - 59T^{2} \)
61 \( 1 + 0.282iT - 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 7.55iT - 79T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 - 7.27iT - 89T^{2} \)
97 \( 1 - 4.07T + 97T^{2} \)
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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

−10.07204335009030140353246117570, −9.335903320854127499705663832046, −8.109154993773101073301605870847, −7.60896966577941365073685013242, −6.94461900226527145867326171796, −5.13634486970056919066464811216, −4.18182145829143048527996615502, −3.69553430900111470133143637831, −2.27870933920699476620791942491, −0.796225710235994027305509237984, 0.833028195587337107124109422426, 2.96801217325432331971456842772, 4.03916693590014856237252692680, 5.23429114203485034991971665218, 5.89617199883525669182211711385, 6.87630470459564015562753974148, 7.82897286796738508569044261835, 8.459931829603442517075313237770, 9.005615205978507548970944713345, 9.995926638666296909487268425429

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