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Properties

Label 2-2368-37.22-c0-0-0
Degree $2$
Conductor $2368$
Sign $0.587 + 0.809i$
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  + (−1.48 + 1.03i)5-s + (0.766 − 0.642i)9-s + (−0.168 − 1.92i)13-s + (−1.63 − 0.142i)17-s + (0.781 − 2.14i)25-s + (1.92 + 0.515i)29-s + (0.642 − 0.766i)37-s + (0.223 − 0.266i)41-s + (−0.469 + 1.75i)45-s + (0.939 + 0.342i)49-s + (−0.300 − 1.70i)53-s + (−0.173 + 0.0151i)61-s + (2.25 + 2.68i)65-s − i·73-s + (0.173 − 0.984i)81-s + ⋯
L(s)  = 1  + (−1.48 + 1.03i)5-s + (0.766 − 0.642i)9-s + (−0.168 − 1.92i)13-s + (−1.63 − 0.142i)17-s + (0.781 − 2.14i)25-s + (1.92 + 0.515i)29-s + (0.642 − 0.766i)37-s + (0.223 − 0.266i)41-s + (−0.469 + 1.75i)45-s + (0.939 + 0.342i)49-s + (−0.300 − 1.70i)53-s + (−0.173 + 0.0151i)61-s + (2.25 + 2.68i)65-s − i·73-s + (0.173 − 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7751042249\)
\(L(\frac12)\) \(\approx\) \(0.7751042249\)
\(L(1)\) 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 \( 1 + (1.48 - 1.03i)T + (0.342 - 0.939i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.168 + 1.92i)T + (-0.984 + 0.173i)T^{2} \)
17 \( 1 + (1.63 + 0.142i)T + (0.984 + 0.173i)T^{2} \)
19 \( 1 + (0.642 + 0.766i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1.92 - 0.515i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.173 - 0.0151i)T + (0.984 - 0.173i)T^{2} \)
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 \( 1 + (0.939 + 0.657i)T + (0.342 + 0.939i)T^{2} \)
97 \( 1 + (0.168 - 0.0451i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.871995805859840484317998409836, −8.156178925540906443147011525723, −7.47005319444501469447586003173, −6.86584177549047738953908589396, −6.19120675228070878555633173352, −4.87420522822005081365565108105, −4.10892713501252284592454507426, −3.30332327734981337179761433294, −2.57448421664288727741478671711, −0.58785151078883826517368542423, 1.28460149461262021991744209716, 2.48294979269105467689289804185, 4.08214922956590154565274668954, 4.39117028296220537265068878953, 4.88163031752523660185308691712, 6.40174013462479410502435679846, 7.05402746495686245821476209064, 7.79264878159805156582240925446, 8.587184641539569494054459959773, 9.008279625171101980235636261169

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