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Properties

Label 2-2352-147.65-c0-0-0
Degree $2$
Conductor $2352$
Sign $-0.814 + 0.580i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0747 − 0.997i)3-s + (−0.365 − 0.930i)7-s + (−0.988 + 0.149i)9-s + (0.0931 − 0.116i)13-s + (0.365 − 0.632i)19-s + (−0.900 + 0.433i)21-s + (0.365 − 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (−1.40 − 1.29i)37-s + (−0.123 − 0.0841i)39-s + (−1.78 + 0.858i)43-s + (−0.733 + 0.680i)49-s + (−0.658 − 0.317i)57-s + (0.326 + 0.302i)61-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)3-s + (−0.365 − 0.930i)7-s + (−0.988 + 0.149i)9-s + (0.0931 − 0.116i)13-s + (0.365 − 0.632i)19-s + (−0.900 + 0.433i)21-s + (0.365 − 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (−1.40 − 1.29i)37-s + (−0.123 − 0.0841i)39-s + (−1.78 + 0.858i)43-s + (−0.733 + 0.680i)49-s + (−0.658 − 0.317i)57-s + (0.326 + 0.302i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8551661228\)
\(L(\frac12)\) \(\approx\) \(0.8551661228\)
\(L(1)\) 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 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (0.365 + 0.930i)T \)
good5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.78 - 0.858i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-0.326 - 0.302i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.733 - 1.26i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−8.719373301755938366123871317273, −7.970657010786165553963179515404, −7.21839624300453486578930546660, −6.72870569723742486906434895542, −5.89437948154302439856930705832, −4.98705602230301195426840108180, −3.88951260804115629707251228425, −2.96370648453313589758515948768, −1.86285574829364043728374985087, −0.56987836057923689590271086419, 1.84097077124193558520606011269, 3.18796396727950959929257972662, 3.57944145828076086601548165902, 4.98334127876693639700209609174, 5.30198579000323545210160062595, 6.26468386380332871275093496671, 7.05054575873653657031738137107, 8.306752472761293131041532461043, 8.749880039471963261096054220503, 9.493090790697534082215187820111

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