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Properties

Label 2-2368-37.6-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.646 - 0.763i$
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  + i·3-s + (1 + i)5-s − 7-s + i·11-s + (−1 − i)13-s + (−1 + i)15-s − i·21-s + (1 + i)23-s + i·25-s + i·27-s + (−1 + i)29-s − 33-s + (−1 − i)35-s + 37-s + (1 − i)39-s + ⋯
L(s)  = 1  + i·3-s + (1 + i)5-s − 7-s + i·11-s + (−1 − i)13-s + (−1 + i)15-s − i·21-s + (1 + i)23-s + i·25-s + i·27-s + (−1 + i)29-s − 33-s + (−1 − i)35-s + 37-s + (1 − i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.646 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177881353\)
\(L(\frac12)\) \(\approx\) \(1.177881353\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.584884824197490572710310952119, −9.213562927185144590442048276750, −7.70235498441598611791804373852, −7.04252880988148823826537993481, −6.40017030399039791085852546921, −5.36385579259932483480011486118, −4.85127538154988837202089334076, −3.52166644331070202787119388188, −3.02394423742127945405583830380, −1.93536883827544989141878694947, 0.78077145052131253863577370504, 1.92922743326551610108263596566, 2.75089738665443547758264968281, 4.10494113285929801079505401105, 5.06987677481249570320039762999, 5.94200494687919938743555753875, 6.54996800064447569771127628759, 7.16949797029514871679391394019, 8.188994542937906160475285769996, 8.922880485603564792970134179780

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