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Properties

Label 2-936-104.83-c1-0-60
Degree $2$
Conductor $936$
Sign $-0.372 + 0.927i$
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  + (0.915 + 1.07i)2-s + (−0.321 + 1.97i)4-s + (−1.17 − 1.17i)5-s + (−0.308 + 0.308i)7-s + (−2.42 + 1.46i)8-s + (0.189 − 2.33i)10-s + (−3.54 − 3.54i)11-s + (−3.38 + 1.24i)13-s + (−0.614 − 0.0497i)14-s + (−3.79 − 1.27i)16-s − 3.87i·17-s + (−2.34 + 2.34i)19-s + (2.68 − 1.93i)20-s + (0.571 − 7.05i)22-s + 0.632·23-s + ⋯
L(s)  = 1  + (0.647 + 0.761i)2-s + (−0.160 + 0.986i)4-s + (−0.523 − 0.523i)5-s + (−0.116 + 0.116i)7-s + (−0.856 + 0.516i)8-s + (0.0597 − 0.737i)10-s + (−1.06 − 1.06i)11-s + (−0.938 + 0.344i)13-s + (−0.164 − 0.0132i)14-s + (−0.948 − 0.317i)16-s − 0.938i·17-s + (−0.537 + 0.537i)19-s + (0.600 − 0.432i)20-s + (0.121 − 1.50i)22-s + 0.131·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.372 + 0.927i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.372 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112797 - 0.166843i\)
\(L(\frac12)\) \(\approx\) \(0.112797 - 0.166843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.915 - 1.07i)T \)
3 \( 1 \)
13 \( 1 + (3.38 - 1.24i)T \)
good5 \( 1 + (1.17 + 1.17i)T + 5iT^{2} \)
7 \( 1 + (0.308 - 0.308i)T - 7iT^{2} \)
11 \( 1 + (3.54 + 3.54i)T + 11iT^{2} \)
17 \( 1 + 3.87iT - 17T^{2} \)
19 \( 1 + (2.34 - 2.34i)T - 19iT^{2} \)
23 \( 1 - 0.632T + 23T^{2} \)
29 \( 1 - 5.86iT - 29T^{2} \)
31 \( 1 + (4.22 + 4.22i)T + 31iT^{2} \)
37 \( 1 + (5.95 - 5.95i)T - 37iT^{2} \)
41 \( 1 + (-6.84 + 6.84i)T - 41iT^{2} \)
43 \( 1 - 3.52iT - 43T^{2} \)
47 \( 1 + (3.45 - 3.45i)T - 47iT^{2} \)
53 \( 1 + 8.09iT - 53T^{2} \)
59 \( 1 + (5.28 + 5.28i)T + 59iT^{2} \)
61 \( 1 - 3.42iT - 61T^{2} \)
67 \( 1 + (-9.72 + 9.72i)T - 67iT^{2} \)
71 \( 1 + (5.43 + 5.43i)T + 71iT^{2} \)
73 \( 1 + (2.76 + 2.76i)T + 73iT^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (8.72 - 8.72i)T - 83iT^{2} \)
89 \( 1 + (-3.17 - 3.17i)T + 89iT^{2} \)
97 \( 1 + (-1.60 + 1.60i)T - 97iT^{2} \)
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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

−9.561535702418914316062239269265, −8.699964463069406509780947391512, −7.994106837000862900703624128805, −7.31369628937186209160802491643, −6.28790646326524743466405468145, −5.30044416401926905951829858369, −4.71086982037383258928443488667, −3.56140922367208687488165390015, −2.55409696544325780719777900375, −0.06977318447164659251556139490, 2.01373278635027230815672196576, 2.92221187872318316038895197099, 4.01670971154118770584409226874, 4.87283473822407857520023462055, 5.77116490409452684603513684536, 6.98044406319105254500889875446, 7.58630117436284496262697664297, 8.807132278291105108093317391311, 9.895126760742064776015912356784, 10.40228522273910871842884198074

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