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Properties

Label 2-2368-37.19-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.775 + 0.631i$
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  + (−0.842 − 1.80i)5-s + (0.173 − 0.984i)9-s + (−0.296 + 0.424i)13-s + (0.692 − 0.484i)17-s + (−1.90 + 2.27i)25-s + (−0.424 − 1.58i)29-s + (0.984 − 0.173i)37-s + (−1.85 + 0.326i)41-s + (−1.92 + 0.515i)45-s + (−0.766 − 0.642i)49-s + (−1.62 + 0.592i)53-s + (0.939 + 0.657i)61-s + (1.01 + 0.179i)65-s − i·73-s + (−0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (−0.842 − 1.80i)5-s + (0.173 − 0.984i)9-s + (−0.296 + 0.424i)13-s + (0.692 − 0.484i)17-s + (−1.90 + 2.27i)25-s + (−0.424 − 1.58i)29-s + (0.984 − 0.173i)37-s + (−1.85 + 0.326i)41-s + (−1.92 + 0.515i)45-s + (−0.766 − 0.642i)49-s + (−1.62 + 0.592i)53-s + (0.939 + 0.657i)61-s + (1.01 + 0.179i)65-s − i·73-s + (−0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8024145865\)
\(L(\frac12)\) \(\approx\) \(0.8024145865\)
\(L(1)\) 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 \( 1 + (0.842 + 1.80i)T + (-0.642 + 0.766i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.296 - 0.424i)T + (-0.342 - 0.939i)T^{2} \)
17 \( 1 + (-0.692 + 0.484i)T + (0.342 - 0.939i)T^{2} \)
19 \( 1 + (0.984 + 0.173i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.424 + 1.58i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.939 - 0.657i)T + (0.342 + 0.939i)T^{2} \)
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 \( 1 + (-0.766 + 1.64i)T + (-0.642 - 0.766i)T^{2} \)
97 \( 1 + (0.296 - 1.10i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.931549305336532893906881121920, −8.075637069256193377255239560579, −7.59682844907475983280290991924, −6.52370036290932488655816624642, −5.58307559384401855837128164596, −4.76320457828550975828552467529, −4.14173331591447609122868730130, −3.30480369698167129879669665493, −1.66505548420690185479203053987, −0.54731596265860863126071136854, 1.92766153622919029303559094208, 3.01957060518417347362801710643, 3.53024369561038903400548358641, 4.63951712988266464352079497117, 5.60175226344822948182057186573, 6.60716496759338018191124383106, 7.16807036224602910185502711336, 7.88514218915862975807065317371, 8.332624566255509495227144507422, 9.772963796741080743865404620737

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