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Properties

Label 2-2368-148.11-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.729 - 0.683i$
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  + (−1.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1.5 + 0.866i)17-s + (1 + 1.73i)25-s + 1.73i·29-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + 1.73i·45-s + (−0.5 − 0.866i)49-s + (−1 − 1.73i)53-s + (1.5 + 0.866i)61-s − 2·73-s + (−0.499 + 0.866i)81-s + 3·85-s + (−1.5 + 0.866i)89-s + ⋯
L(s)  = 1  + (−1.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1.5 + 0.866i)17-s + (1 + 1.73i)25-s + 1.73i·29-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + 1.73i·45-s + (−0.5 − 0.866i)49-s + (−1 − 1.73i)53-s + (1.5 + 0.866i)61-s − 2·73-s + (−0.499 + 0.866i)81-s + 3·85-s + (−1.5 + 0.866i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.729 - 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07270820257\)
\(L(\frac12)\) \(\approx\) \(0.07270820257\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.164328376859079429387741497273, −8.415420263564868456884686396041, −8.363174655410042928752370879835, −7.06227398772007368315044391386, −6.57536128373172578938049668228, −5.37449270022707896986591329571, −4.57713386096475886614147249679, −3.83773075808683546716801186222, −3.10249358751689742005302207103, −1.46373493786994831767566553048, 0.04895979840708771525429141232, 2.28408499071907358303015949912, 3.00487299723561968994339649396, 4.09165613931266401388875251230, 4.64717663737698816033008978405, 5.79024389814455139125270959054, 6.78474125423385825986967435555, 7.37043037320151932099873330955, 8.024696657792606228288137003021, 8.670176969286064619805323619187

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