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Properties

Label 2-936-104.101-c1-0-26
Degree $2$
Conductor $936$
Sign $0.432 - 0.901i$
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  + (−1.03 + 0.967i)2-s + (0.128 − 1.99i)4-s + 1.39·5-s + (2.60 + 1.50i)7-s + (1.79 + 2.18i)8-s + (−1.43 + 1.34i)10-s + (1.67 + 2.90i)11-s + (3.25 − 1.56i)13-s + (−4.14 + 0.969i)14-s + (−3.96 − 0.512i)16-s + (−1.30 + 2.25i)17-s + (1.73 − 3.01i)19-s + (0.179 − 2.78i)20-s + (−4.54 − 1.37i)22-s + (0.413 + 0.716i)23-s + ⋯
L(s)  = 1  + (−0.729 + 0.684i)2-s + (0.0642 − 0.997i)4-s + 0.623·5-s + (0.985 + 0.568i)7-s + (0.635 + 0.771i)8-s + (−0.454 + 0.426i)10-s + (0.505 + 0.876i)11-s + (0.901 − 0.432i)13-s + (−1.10 + 0.259i)14-s + (−0.991 − 0.128i)16-s + (−0.315 + 0.546i)17-s + (0.398 − 0.690i)19-s + (0.0400 − 0.622i)20-s + (−0.968 − 0.293i)22-s + (0.0862 + 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.432 - 0.901i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.432 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22722 + 0.772700i\)
\(L(\frac12)\) \(\approx\) \(1.22722 + 0.772700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 - 0.967i)T \)
3 \( 1 \)
13 \( 1 + (-3.25 + 1.56i)T \)
good5 \( 1 - 1.39T + 5T^{2} \)
7 \( 1 + (-2.60 - 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.30 - 2.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.413 - 0.716i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.52 - 0.878i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.22iT - 31T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.270 - 0.156i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.10 - 3.52i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.27iT - 47T^{2} \)
53 \( 1 - 0.507iT - 53T^{2} \)
59 \( 1 + (-6.95 + 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.47 - 4.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.361 - 0.626i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.51 + 2.60i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + 1.78T + 79T^{2} \)
83 \( 1 - 7.28T + 83T^{2} \)
89 \( 1 + (10.3 - 5.94i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.96 + 1.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.922413087352174737717892718422, −9.310794156663165063988801499178, −8.530330877276091726399189729298, −7.80489162154714542447769297365, −6.85708665675524480451944998430, −5.93036142080295542952453976039, −5.27957719155866462033798092374, −4.21012852635437344344007582046, −2.28835443459805587492823705942, −1.36231401709805390777230926909, 1.05459380799982056969983586469, 2.00171638771209459088757066240, 3.43688998999826490205453979172, 4.28887623431250254030454809199, 5.57926981325031211828084864207, 6.66309573851332121677432781869, 7.59130545524937095175011105559, 8.493300189621760599360817503966, 9.023478322613132339559483510152, 10.00806578550797771900026513296

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