Properties

Label 2-2352-12.11-c1-0-18
Degree $2$
Conductor $2352$
Sign $-i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  − 1.73i·3-s + 3.74i·5-s − 2.99·9-s + 6.48·11-s + 6.48·15-s + 3.74i·17-s + 6.92i·19-s − 6.48·23-s − 9·25-s + 5.19i·27-s − 3.46i·31-s − 11.2i·33-s − 8·37-s − 3.74i·41-s − 11.2i·45-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.67i·5-s − 0.999·9-s + 1.95·11-s + 1.67·15-s + 0.907i·17-s + 1.58i·19-s − 1.35·23-s − 1.80·25-s + 0.999i·27-s − 0.622i·31-s − 1.95i·33-s − 1.31·37-s − 0.584i·41-s − 1.67i·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417147706\)
\(L(\frac12)\) \(\approx\) \(1.417147706\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 - 3.74iT - 5T^{2} \)
11 \( 1 - 6.48T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3.74iT - 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + 6.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 3.74iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 6.48T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.7iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.100678151218097145769525664544, −8.187853372965846273578686357856, −7.54862077601046934854046130560, −6.71454181234570373014540900870, −6.29706260960831750613270686979, −5.75108511108952738878256909680, −3.85710675981420994398508145602, −3.57551004297520296104969769845, −2.23624399118433364632000541698, −1.52049323604811346340513537758, 0.48232553811209032471025234907, 1.70430002482312423843975471442, 3.21533409493816358890282660618, 4.23988108097831991687896551062, 4.61877063343230696467379275239, 5.41961881032805411772378256416, 6.29291272895179281096972381301, 7.26637414522952538136769554892, 8.621100075088351538587801062873, 8.726579082038831138600136872055

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