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Properties

Label 2-936-104.77-c1-0-43
Degree $2$
Conductor $936$
Sign $0.993 - 0.111i$
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  + (0.831 + 1.14i)2-s + (−0.618 + 1.90i)4-s − 2.68·5-s − 4.15i·7-s + (−2.68 + 0.874i)8-s + (−2.23 − 3.07i)10-s + 4.35·11-s + (−3.53 − 0.726i)13-s + (4.74 − 3.45i)14-s + (−3.23 − 2.35i)16-s + 5.87·17-s + 5.71·19-s + (1.66 − 5.11i)20-s + (3.61 + 4.97i)22-s + 3.62·23-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s − 1.20·5-s − 1.56i·7-s + (−0.951 + 0.309i)8-s + (−0.707 − 0.973i)10-s + 1.31·11-s + (−0.979 − 0.201i)13-s + (1.26 − 0.922i)14-s + (−0.809 − 0.587i)16-s + 1.42·17-s + 1.31·19-s + (0.371 − 1.14i)20-s + (0.771 + 1.06i)22-s + 0.756·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62427 + 0.0904542i\)
\(L(\frac12)\) \(\approx\) \(1.62427 + 0.0904542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 - 1.14i)T \)
3 \( 1 \)
13 \( 1 + (3.53 + 0.726i)T \)
good5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
17 \( 1 - 5.87T + 17T^{2} \)
19 \( 1 - 5.71T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 - 3.08iT - 29T^{2} \)
31 \( 1 + 9.28iT - 31T^{2} \)
37 \( 1 - 2.69T + 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + 3.80iT - 43T^{2} \)
47 \( 1 - 4.91iT - 47T^{2} \)
53 \( 1 + 1.17iT - 53T^{2} \)
59 \( 1 - 2.29T + 59T^{2} \)
61 \( 1 - 7.05iT - 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + 5.13iT - 97T^{2} \)
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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

−9.954657034769944898066435536156, −9.142609679925739479673480349550, −7.896465405489910252029715625436, −7.39820818198279096911359474723, −7.04077954855975919599321300415, −5.71454441076596305059911783905, −4.61708528273238895045712193734, −3.87204110562647381608084229858, −3.28117465060776990420451871659, −0.75396258625212367195181762041, 1.29680534048491174716700405145, 2.84098285918952507367246902747, 3.50456771676658173629144663505, 4.69891210628850477122264134461, 5.41381801428176963799553612283, 6.44770691655238150154524833215, 7.53820353064071321247667691023, 8.579913582881314237509787632353, 9.385222339808840414310945757848, 9.945743030251585874836743180974

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