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Properties

Label 2-2368-148.99-c0-0-0
Degree $2$
Conductor $2368$
Sign $0.880 + 0.473i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  + (−0.439 − 1.20i)5-s + (0.173 + 0.984i)9-s + (1.70 + 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.500 + 0.419i)25-s + (−1.70 + 0.984i)29-s + (0.173 − 0.984i)37-s + (0.326 − 1.85i)41-s + (1.11 − 0.642i)45-s + (0.766 − 0.642i)49-s + (0.939 + 0.342i)53-s + (1.93 + 0.342i)61-s + (−0.386 − 2.19i)65-s + 73-s + (−0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (−0.439 − 1.20i)5-s + (0.173 + 0.984i)9-s + (1.70 + 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.500 + 0.419i)25-s + (−1.70 + 0.984i)29-s + (0.173 − 0.984i)37-s + (0.326 − 1.85i)41-s + (1.11 − 0.642i)45-s + (0.766 − 0.642i)49-s + (0.939 + 0.342i)53-s + (1.93 + 0.342i)61-s + (−0.386 − 2.19i)65-s + 73-s + (−0.939 + 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214105740\)
\(L(\frac12)\) \(\approx\) \(1.214105740\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-0.173 + 0.984i)T \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
19 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.815635383092499452029239879197, −8.536088080521577606382041542873, −7.65562727317272596393228983400, −6.96509161040804269730743367030, −5.56862482825686354312267256082, −5.39388470920812076242141129273, −4.14243851640020643207620660833, −3.69430585219610083031810551924, −2.11159898164866668605803998866, −1.07613740512076809580628746956, 1.20794125716447073150043519404, 2.73386903218811606384987542532, 3.61202208343498210630693230577, 3.99310919285542694117527609259, 5.52828666431204733161507465368, 6.26580234152898598346036334100, 6.78726167673518603195833210821, 7.72074156131697331383886748350, 8.315901732106205254446448451119, 9.305937855954328915615383763615

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