Properties

Label 2-2352-28.27-c1-0-3
Degree $2$
Conductor $2352$
Sign $-0.944 - 0.327i$
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  − 3-s + 1.73i·5-s + 9-s + 5.19i·11-s + 6.92i·13-s − 1.73i·15-s + 3.46i·17-s + 2·19-s − 6.92i·23-s + 2.00·25-s − 27-s − 9·29-s + 31-s − 5.19i·33-s − 2·37-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.774i·5-s + 0.333·9-s + 1.56i·11-s + 1.92i·13-s − 0.447i·15-s + 0.840i·17-s + 0.458·19-s − 1.44i·23-s + 0.400·25-s − 0.192·27-s − 1.67·29-s + 0.179·31-s − 0.904i·33-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9797051281\)
\(L(\frac12)\) \(\approx\) \(0.9797051281\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−9.399313986225159804161803806916, −8.689566754130849929814897972891, −7.44570103384186901012538675461, −6.94111526070505957458140205928, −6.46142402800890392576405098808, −5.41254595532140251450104078766, −4.41899804655451168354070894288, −3.91557220328210265752888404067, −2.41829803001243573852728731649, −1.66366257881429197187179758114, 0.38878806783810151063413324181, 1.24236527673125871696258896508, 2.96422302701552219325839491158, 3.62439317450712667913210896472, 4.95379099453019211167493133182, 5.54110775059362398383594301348, 5.93059242114323604959642214856, 7.23480578719337984755746912612, 7.87714954488260912208687688887, 8.619054464138582699759206127593

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