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Properties

Label 2-2352-12.11-c1-0-53
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 − 2.99·9-s + 7·13-s − 5.19i·19-s + 5·25-s + 5.19i·27-s + 8.66i·31-s − 37-s − 12.1i·39-s − 12.1i·43-s − 9·57-s + 14·61-s − 12.1i·67-s + 7·73-s − 8.66i·75-s + ⋯
L(s)  = 1  − 0.999i·3-s − 0.999·9-s + 1.94·13-s − 1.19i·19-s + 25-s + 0.999i·27-s + 1.55i·31-s − 0.164·37-s − 1.94i·39-s − 1.84i·43-s − 1.19·57-s + 1.79·61-s − 1.48i·67-s + 0.819·73-s − 0.999i·75-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

\(L(1)\) \(\approx\) \(1.823139826\)
\(L(\frac12)\) \(\approx\) \(1.823139826\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.662860358364950484133407084626, −8.170107535664199386557450119400, −6.97566821064394286895211477384, −6.72052141576389241610331893337, −5.76719395592021258873440951344, −4.99599427830733912744583560435, −3.71259627038352938953321614655, −2.90982759903641038627608475283, −1.72507014875236128408896494049, −0.73733607608865045705519489337, 1.16568221806449474142196239590, 2.64893838576950113212914994972, 3.70721667549705952982994994509, 4.12467184281138838191689878433, 5.27960588827974201800863393800, 5.96049951223975620415082524032, 6.62886201628499999644719600026, 8.014814095905459678834855941958, 8.384998978592289281633901685596, 9.235978052144681948681978355269

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