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Properties

Label 2-2352-84.47-c0-0-1
Degree $2$
Conductor $2352$
Sign $-0.0633 - 0.997i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational no
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 + (−0.499 + 0.866i)9-s + (−1 + 1.73i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (1 + 1.73i)31-s + (1 − 1.73i)37-s − 1.99·57-s + (−0.499 + 0.866i)75-s + (−0.5 − 0.866i)81-s + (−0.999 + 1.73i)93-s + (1 − 1.73i)103-s + (−1 − 1.73i)109-s + 1.99·111-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−1 + 1.73i)19-s + (0.5 + 0.866i)25-s − 0.999·27-s + (1 + 1.73i)31-s + (1 − 1.73i)37-s − 1.99·57-s + (−0.499 + 0.866i)75-s + (−0.5 − 0.866i)81-s + (−0.999 + 1.73i)93-s + (1 − 1.73i)103-s + (−1 − 1.73i)109-s + 1.99·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297201511\)
\(L(\frac12)\) \(\approx\) \(1.297201511\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.347913553467959116379971356881, −8.620886641361845600039444231418, −8.059504669325826181063202993505, −7.18155410114895145859927380363, −6.12154568390454536581049972812, −5.37298834629383443464695512859, −4.44880349665391334457367252945, −3.73112614678026613213519520011, −2.85487618404965542030039888056, −1.71918953262364424806938255793, 0.845557552588317244927063311378, 2.29856121296980983520458619846, 2.84971420831768090795136390045, 4.12244356078066379432436476872, 4.92489926098771545132311834879, 6.28431738649007638057939817588, 6.51005196797831603293376521922, 7.51604790775848753725465989904, 8.191822468902443548097144435338, 8.829984375487463093238543542460

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