Properties

Degree $2$
Conductor $2352$
Sign $0.386 + 0.922i$
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 + (1.63 − 2.83i)5-s + (−0.499 + 0.866i)9-s + (1.63 + 2.83i)11-s − 6.27·13-s + 3.27·15-s + (−2 − 3.46i)17-s + (3.13 − 5.43i)19-s + (2 − 3.46i)23-s + (−2.86 − 4.95i)25-s − 0.999·27-s + 5.27·29-s + (0.5 + 0.866i)31-s + (−1.63 + 2.83i)33-s + (1.13 − 1.97i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.732 − 1.26i)5-s + (−0.166 + 0.288i)9-s + (0.493 + 0.855i)11-s − 1.74·13-s + 0.845·15-s + (−0.485 − 0.840i)17-s + (0.719 − 1.24i)19-s + (0.417 − 0.722i)23-s + (−0.572 − 0.991i)25-s − 0.192·27-s + 0.979·29-s + (0.0898 + 0.155i)31-s + (−0.285 + 0.493i)33-s + (0.186 − 0.323i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956521717\)
\(L(\frac12)\) \(\approx\) \(1.956521717\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 2.83i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.13 + 5.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.13 + 1.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.63 + 8.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.137 + 0.238i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (2.13 + 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.77 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + (5.27 - 9.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−9.176196928244880560691463045976, −8.255338682953251800649173205736, −7.25856706610758724539215818904, −6.64064379143275012134430479098, −5.23542940510970871024279005429, −4.91418258126511507797199753123, −4.34738368378736158257136114385, −2.81019490162364160959297187420, −2.08236311795215321119045462397, −0.64162726580858511342487884369, 1.38066338343555960813660473771, 2.50867186705510096959058133403, 3.07005034339211272080342764424, 4.14664757588303007830343285264, 5.47751711622525871773557786241, 6.12589913968863992894977274003, 6.81813008342322311042343838560, 7.50339292998233968638896067729, 8.213146317833152790443524636586, 9.264402194909567498153410827578

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