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Properties

Label 2-1368-152.75-c1-0-9
Degree $2$
Conductor $1368$
Sign $-0.999 - 0.00742i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (−1.17 − 0.792i)2-s + (0.743 + 1.85i)4-s + 3.51i·5-s + 2.23i·7-s + (0.601 − 2.76i)8-s + (2.78 − 4.12i)10-s − 2.89·11-s − 6.30·13-s + (1.77 − 2.61i)14-s + (−2.89 + 2.76i)16-s + 4.79·17-s + (0.895 + 4.26i)19-s + (−6.53 + 2.61i)20-s + (3.39 + 2.29i)22-s − 0.524i·23-s + ⋯
L(s)  = 1  + (−0.828 − 0.560i)2-s + (0.371 + 0.928i)4-s + 1.57i·5-s + 0.845i·7-s + (0.212 − 0.977i)8-s + (0.882 − 1.30i)10-s − 0.872·11-s − 1.74·13-s + (0.473 − 0.699i)14-s + (−0.723 + 0.690i)16-s + 1.16·17-s + (0.205 + 0.978i)19-s + (−1.46 + 0.584i)20-s + (0.722 + 0.489i)22-s − 0.109i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.999 - 0.00742i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.999 - 0.00742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4011846349\)
\(L(\frac12)\) \(\approx\) \(0.4011846349\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.792i)T \)
3 \( 1 \)
19 \( 1 + (-0.895 - 4.26i)T \)
good5 \( 1 - 3.51iT - 5T^{2} \)
7 \( 1 - 2.23iT - 7T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 + 6.30T + 13T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
23 \( 1 + 0.524iT - 23T^{2} \)
29 \( 1 + 0.415T + 29T^{2} \)
31 \( 1 - 1.20T + 31T^{2} \)
37 \( 1 - 5.88T + 37T^{2} \)
41 \( 1 - 4.87iT - 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 4.27iT - 47T^{2} \)
53 \( 1 + 6.05T + 53T^{2} \)
59 \( 1 + 8.08iT - 59T^{2} \)
61 \( 1 + 8.38iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 7.76T + 73T^{2} \)
79 \( 1 + 7.33T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 - 2.19iT - 97T^{2} \)
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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

−9.812711155063105192081091902060, −9.717941560441764809384535911624, −8.186351693789061593808353522357, −7.74148647739352282975304243639, −6.98688454765908178401280444974, −6.07130154802146653228354083067, −4.97907055836120784278100296892, −3.41075304839107251022439885606, −2.79695399464681005714834771590, −2.02996410790628252407388626314, 0.22263763420318274154589110549, 1.25598909675237053382991635303, 2.67112985830506766934621620222, 4.47826941184671968773460304765, 5.05001732832748769147471241531, 5.73905844168679824348969132474, 7.16718840278090359441421822547, 7.57893963594441193460320650052, 8.338141435599740536039460809529, 9.168839396666341257050275013392

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