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Properties

Label 2-936-8.5-c1-0-48
Degree $2$
Conductor $936$
Sign $-0.509 + 0.860i$
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  + (0.913 − 1.07i)2-s + (−0.332 − 1.97i)4-s − 0.592i·5-s + 3.54·7-s + (−2.43 − 1.44i)8-s + (−0.639 − 0.540i)10-s − 2.11i·11-s − i·13-s + (3.24 − 3.83i)14-s + (−3.77 + 1.31i)16-s − 1.91·17-s − 2.26i·19-s + (−1.16 + 0.196i)20-s + (−2.28 − 1.93i)22-s + 1.43·23-s + ⋯
L(s)  = 1  + (0.645 − 0.763i)2-s + (−0.166 − 0.986i)4-s − 0.264i·5-s + 1.34·7-s + (−0.860 − 0.509i)8-s + (−0.202 − 0.171i)10-s − 0.637i·11-s − 0.277i·13-s + (0.866 − 1.02i)14-s + (−0.944 + 0.327i)16-s − 0.463·17-s − 0.518i·19-s + (−0.261 + 0.0440i)20-s + (−0.486 − 0.411i)22-s + 0.299·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15899 - 2.03371i\)
\(L(\frac12)\) \(\approx\) \(1.15899 - 2.03371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.913 + 1.07i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 0.592iT - 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 + 2.11iT - 11T^{2} \)
17 \( 1 + 1.91T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 + 0.214iT - 29T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 + 3.57iT - 37T^{2} \)
41 \( 1 + 0.377T + 41T^{2} \)
43 \( 1 + 0.319iT - 43T^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 + 9.40iT - 53T^{2} \)
59 \( 1 + 2.37iT - 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 - 6.49iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 3.57T + 73T^{2} \)
79 \( 1 - 4.55T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−10.01138800363008111524436882597, −8.897673583661552844522652966056, −8.383018532446776363516403966720, −7.16053159506704180324928042020, −6.05341677188260840207374008865, −5.07133885674250304033182551096, −4.56613911394159424133150966468, −3.34198509846568206202898570743, −2.17392364516266091928964770757, −0.955845712134721712839739561880, 1.85976020936456851605892587544, 3.17919875560143413424684983048, 4.51953327194965736852077727168, 4.87720022104636506754615163297, 6.06276195591203368825307876308, 6.93872645310366537645127217259, 7.71566034182092227269095821017, 8.417354486686078604856553476750, 9.253993374515846193128291547960, 10.46196807864283646313953610135

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