Properties

Degree $2$
Conductor $2352$
Sign $-0.900 + 0.435i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (1.70 − 2.95i)5-s + (−0.499 + 0.866i)9-s + (2.41 + 4.18i)11-s − 1.41·13-s − 3.41·15-s + (−3.12 − 5.40i)17-s + (−0.585 + 1.01i)19-s + (−0.414 + 0.717i)23-s + (−3.32 − 5.76i)25-s + 0.999·27-s − 8.48·29-s + (−5.41 − 9.37i)31-s + (2.41 − 4.18i)33-s + (4.82 − 8.36i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.763 − 1.32i)5-s + (−0.166 + 0.288i)9-s + (0.727 + 1.26i)11-s − 0.392·13-s − 0.881·15-s + (−0.757 − 1.31i)17-s + (−0.134 + 0.232i)19-s + (−0.0863 + 0.149i)23-s + (−0.665 − 1.15i)25-s + 0.192·27-s − 1.57·29-s + (−0.972 − 1.68i)31-s + (0.420 − 0.727i)33-s + (0.793 − 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.900 + 0.435i$
Motivic weight: \(1\)
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238113653\)
\(L(\frac12)\) \(\approx\) \(1.238113653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.41 - 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (3.12 + 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.585 - 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.414 - 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (5.41 + 9.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.82 + 8.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (0.585 - 1.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.65 + 8.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.94 - 5.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 + (-1.53 - 2.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (-7.36 + 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.932389493341205737205929694597, −7.63589739649530108831281401941, −7.30039774233655610312432900340, −6.18213052107709928568859641325, −5.53071345208376988329340464057, −4.72415859287628441558232217898, −4.06784138651864694687367080439, −2.29171940968757990587790912383, −1.73503280561763342349569219973, −0.41350071560858887148526125659, 1.58544750883032014644645527774, 2.78405286191202133044887024297, 3.52451004385857063902016839595, 4.42843307425872158627013801696, 5.71046951547934159512374800317, 6.12281696972679269027015061820, 6.77327552214052816739046429422, 7.67825483624831164410697415493, 8.843366966619785574275387106391, 9.235738536299709006534482469911

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