Properties

Label 2-2368-37.35-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.415 - 0.909i$
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  + (−0.157 − 0.0736i)5-s + (0.173 + 0.984i)9-s + (−1.58 + 1.10i)13-s + (−1.03 + 1.48i)17-s + (−0.623 − 0.742i)25-s + (−1.10 − 0.296i)29-s + (−0.984 − 0.173i)37-s + (1.85 + 0.326i)41-s + (0.0451 − 0.168i)45-s + (−0.766 + 0.642i)49-s + (1.62 + 0.592i)53-s + (0.939 + 1.34i)61-s + (0.331 − 0.0584i)65-s i·73-s + (−0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (−0.157 − 0.0736i)5-s + (0.173 + 0.984i)9-s + (−1.58 + 1.10i)13-s + (−1.03 + 1.48i)17-s + (−0.623 − 0.742i)25-s + (−1.10 − 0.296i)29-s + (−0.984 − 0.173i)37-s + (1.85 + 0.326i)41-s + (0.0451 − 0.168i)45-s + (−0.766 + 0.642i)49-s + (1.62 + 0.592i)53-s + (0.939 + 1.34i)61-s + (0.331 − 0.0584i)65-s i·73-s + (−0.939 + 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.415 - 0.909i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.415 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7145325090\)
\(L(\frac12)\) \(\approx\) \(0.7145325090\)
\(L(1)\) 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 \( 1 + (0.984 + 0.173i)T \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (0.157 + 0.0736i)T + (0.642 + 0.766i)T^{2} \)
7 \( 1 + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.58 - 1.10i)T + (0.342 - 0.939i)T^{2} \)
17 \( 1 + (1.03 - 1.48i)T + (-0.342 - 0.939i)T^{2} \)
19 \( 1 + (-0.984 + 0.173i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.62 - 0.592i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.642 + 0.766i)T^{2} \)
61 \( 1 + (-0.939 - 1.34i)T + (-0.342 + 0.939i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.173 + 0.984i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (0.642 + 0.766i)T^{2} \)
83 \( 1 + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.766 + 0.357i)T + (0.642 - 0.766i)T^{2} \)
97 \( 1 + (1.58 - 0.424i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.365062730256764751405266718788, −8.657525997724018845409445951989, −7.77057773833403543936238425051, −7.24374362449134579898947275202, −6.34197194225964359021058415473, −5.45006361177624774383761267135, −4.45296274417476744122050653137, −4.06002146082532564179215497414, −2.44623267718878885132161866213, −1.89288331769101217720746387497, 0.44670378704311716097396423372, 2.19338486210568794898765827537, 3.10255263613327449160346432806, 4.02451230241464053603866171400, 5.05777305185521914801775970705, 5.63369630131275680231570816500, 6.90896633161421944584221304613, 7.19796363016673693292402816539, 8.072445562655881999449590364026, 9.144898081466869281087855421253

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