Properties

Label 2-2352-28.27-c1-0-16
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 − 3.46i·5-s + 9-s + 3.46i·11-s + 5.19i·13-s − 3.46i·15-s + 6.92i·17-s + 7·19-s − 6.99·25-s + 27-s + 5·31-s + 3.46i·33-s + 37-s + 5.19i·39-s + 10.3i·41-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.54i·5-s + 0.333·9-s + 1.04i·11-s + 1.44i·13-s − 0.894i·15-s + 1.68i·17-s + 1.60·19-s − 1.39·25-s + 0.192·27-s + 0.898·31-s + 0.603i·33-s + 0.164·37-s + 0.832i·39-s + 1.62i·41-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

\(L(1)\) \(\approx\) \(2.229970676\)
\(L(\frac12)\) \(\approx\) \(2.229970676\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + 15.5iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−9.077098149816605804689322185896, −8.283428327816373936012536285578, −7.74342575374860073420515928189, −6.76063285165682888701317490390, −5.86504328037294546805244302341, −4.65581847833894122738803872796, −4.50231487966757216952824292645, −3.36883498192403018895314956568, −1.90290218849118131835966266239, −1.30659169085529516462844336584, 0.77461384771567403970559760467, 2.60776993091578251620407238103, 3.02103679708751992722389490520, 3.65323483423099570485081015706, 5.14094921473163246586876414712, 5.81101484425156099851919297538, 6.84519359952269766687427558868, 7.40215846203377569347344575227, 8.023827692313213056297378893424, 8.925084880993530495395330051953

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