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Properties

Label 2-2368-37.36-c1-0-37
Degree $2$
Conductor $2368$
Sign $0.927 - 0.373i$
Analytic cond. $18.9085$
Root an. cond. $4.34839$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.493·3-s − 1.49i·5-s + 4.26·7-s − 2.75·9-s + 5.13·11-s + 6.13i·13-s − 0.737i·15-s − 4.36i·17-s + 6.36i·19-s + 2.10·21-s + 4.75i·23-s + 2.76·25-s − 2.84·27-s + 6.29i·29-s + 0.873i·31-s + ⋯
L(s)  = 1  + 0.285·3-s − 0.668i·5-s + 1.61·7-s − 0.918·9-s + 1.54·11-s + 1.70i·13-s − 0.190i·15-s − 1.05i·17-s + 1.46i·19-s + 0.459·21-s + 0.991i·23-s + 0.553·25-s − 0.547·27-s + 1.16i·29-s + 0.156i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(18.9085\)
Root analytic conductor: \(4.34839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.456219364\)
\(L(\frac12)\) \(\approx\) \(2.456219364\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.64 - 2.27i)T \)
good3 \( 1 - 0.493T + 3T^{2} \)
5 \( 1 + 1.49iT - 5T^{2} \)
7 \( 1 - 4.26T + 7T^{2} \)
11 \( 1 - 5.13T + 11T^{2} \)
13 \( 1 - 6.13iT - 13T^{2} \)
17 \( 1 + 4.36iT - 17T^{2} \)
19 \( 1 - 6.36iT - 19T^{2} \)
23 \( 1 - 4.75iT - 23T^{2} \)
29 \( 1 - 6.29iT - 29T^{2} \)
31 \( 1 - 0.873iT - 31T^{2} \)
41 \( 1 - 5.87T + 41T^{2} \)
43 \( 1 + 5.37iT - 43T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 - 4.62T + 53T^{2} \)
59 \( 1 + 2.46iT - 59T^{2} \)
61 \( 1 - 6.40iT - 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 8.47T + 71T^{2} \)
73 \( 1 + 8.95T + 73T^{2} \)
79 \( 1 + 16.1iT - 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + 1.22iT - 97T^{2} \)
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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

−8.962099735953969456402741335218, −8.490599075971044350354454169708, −7.55858139186867363269444100120, −6.81907307524831402526439846091, −5.76857741393199882754711848983, −4.97014486764544029288011215342, −4.29257066197420579644222015861, −3.40012083363380122023474813896, −1.84509399882985802839890303996, −1.37273658200530327006317926266, 0.901669866613780092782680736561, 2.20439868525210827322063805099, 3.04500402029554591861422216463, 4.06846203959109591555334659538, 4.94129828168280464072473733336, 5.84656395121204590912668231225, 6.58151548890386197570577964271, 7.53891131309781813925732358287, 8.324467191019490362845481349630, 8.628074041191925973018491594503

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