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Properties

Label 2-2368-37.15-c0-0-0
Degree $2$
Conductor $2368$
Sign $0.999 - 0.0403i$
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  + (0.484 + 0.692i)5-s + (0.766 − 0.642i)9-s + (0.515 − 0.0451i)13-s + (0.0999 − 1.14i)17-s + (0.0976 − 0.268i)25-s + (−0.0451 + 0.168i)29-s + (−0.642 + 0.766i)37-s + (−0.223 + 0.266i)41-s + (0.816 + 0.218i)45-s + (0.939 + 0.342i)49-s + (0.300 + 1.70i)53-s + (−0.173 − 1.98i)61-s + (0.281 + 0.335i)65-s + i·73-s + (0.173 − 0.984i)81-s + ⋯
L(s)  = 1  + (0.484 + 0.692i)5-s + (0.766 − 0.642i)9-s + (0.515 − 0.0451i)13-s + (0.0999 − 1.14i)17-s + (0.0976 − 0.268i)25-s + (−0.0451 + 0.168i)29-s + (−0.642 + 0.766i)37-s + (−0.223 + 0.266i)41-s + (0.816 + 0.218i)45-s + (0.939 + 0.342i)49-s + (0.300 + 1.70i)53-s + (−0.173 − 1.98i)61-s + (0.281 + 0.335i)65-s + i·73-s + (0.173 − 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.999 - 0.0403i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.999 - 0.0403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421020025\)
\(L(\frac12)\) \(\approx\) \(1.421020025\)
\(L(1)\) 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 \( 1 + (0.642 - 0.766i)T \)
good3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (-0.484 - 0.692i)T + (-0.342 + 0.939i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.515 + 0.0451i)T + (0.984 - 0.173i)T^{2} \)
17 \( 1 + (-0.0999 + 1.14i)T + (-0.984 - 0.173i)T^{2} \)
19 \( 1 + (-0.642 - 0.766i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.0451 - 0.168i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.300 - 1.70i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.342 + 0.939i)T^{2} \)
61 \( 1 + (0.173 + 1.98i)T + (-0.984 + 0.173i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.766 - 0.642i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (-0.342 + 0.939i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.939 - 1.34i)T + (-0.342 - 0.939i)T^{2} \)
97 \( 1 + (-0.515 - 1.92i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.324053121229672206850665608812, −8.455138551677853827529409055117, −7.46379878955133138676099847574, −6.80407978471759793816163491032, −6.23395475225104996749140299254, −5.27303268002238947615065805269, −4.31569756671060241784573499219, −3.36108068119157241490807934330, −2.50349800453944293371675378255, −1.21895268487049224835683627339, 1.34715716934769137905755898150, 2.13952045297328911126565121617, 3.58491386779741985007068953938, 4.36884333868160965639465437411, 5.27952104995511804198810763171, 5.90105370588280006009442477138, 6.88856244655880334084458076919, 7.64929720599863455652514282277, 8.542839710999900933982236659869, 9.001800337537666304091274388730

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