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Properties

Label 2-2352-7.4-c1-0-24
Degree $2$
Conductor $2352$
Sign $-0.701 + 0.712i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s − 2·13-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + 0.999·27-s + 6·29-s + (−4 + 6.92i)31-s + (−3 − 5.19i)33-s + (−1 − 1.73i)37-s + (1 − 1.73i)39-s − 12·41-s + 4·43-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s − 0.554·13-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + 1.11·29-s + (−0.718 + 1.24i)31-s + (−0.522 − 0.904i)33-s + (−0.164 − 0.284i)37-s + (0.160 − 0.277i)39-s − 1.87·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.646831787816903824132726586090, −8.001374879259783565186792039005, −7.05600172023507380875831306940, −6.48153846101342545074534177586, −5.19296748933218413911097647765, −4.90656848857707710336417235796, −3.93870317497580443598766352268, −2.80677800842291093127547084844, −1.83020412214016604043834328167, 0, 1.27726399541645043448534805157, 2.66195659905419859346654672789, 3.33063664836859835367763984276, 4.65757354639730831547858843281, 5.51503449841269118031303061949, 6.01013707793015992559131022600, 7.07481749841635427497603873488, 7.68051665998343196354861182960, 8.412982407572827495968691533416

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