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Properties

Label 2-936-104.77-c1-0-48
Degree $2$
Conductor $936$
Sign $0.869 + 0.494i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.869 + 0.494i$
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: \((2,\ 936,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24468 - 0.329150i\)
\(L(\frac12)\) \(\approx\) \(1.24468 - 0.329150i\)
\(L(\frac{3}{2})\) 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

−9.967484107616127292985106391267, −9.252017449179354124684384462918, −8.086830691919261055208023288991, −7.59496763038452666709878050197, −6.54371062261716312455608168518, −5.87371637289237688525466787039, −4.98513632046713885797941520910, −3.83019144736018986309873651288, −2.08149480793782563679378734611, −0.77208818447612511145700257561, 1.53457463252706229418495105094, 2.51773224616172865581105152444, 3.27456900392638577592021213609, 5.06578665642444481995057418444, 5.62121853249639021566933447231, 6.67440003220244491506050539103, 8.086958798372005432191747786331, 8.615202921915869167220239675865, 9.317356182076007401826634058484, 10.24114423185826784687675431660

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