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Properties

Label 2-1368-152.75-c1-0-93
Degree $2$
Conductor $1368$
Sign $-i$
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  − 1.41i·2-s − 2.00·4-s − 4.32i·5-s + 2.82i·8-s − 6.11·10-s − 4.15·13-s + 4.00·16-s − 4.35·19-s + 8.65i·20-s − 1.55i·23-s − 13.7·25-s + 5.87i·26-s + 10.2·31-s − 5.65i·32-s − 8.07·37-s + 6.16i·38-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 1.93i·5-s + 1.00i·8-s − 1.93·10-s − 1.15·13-s + 1.00·16-s − 1.00·19-s + 1.93i·20-s − 0.323i·23-s − 2.74·25-s + 1.15i·26-s + 1.84·31-s − 1.00i·32-s − 1.32·37-s + 1.00i·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-i$
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: \((2,\ 1368,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4562519596\)
\(L(\frac12)\) \(\approx\) \(0.4562519596\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
19 \( 1 + 4.35T \)
good5 \( 1 + 4.32iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 1.55iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + 8.71T + 43T^{2} \)
47 \( 1 + 7.10iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8.71T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.980887530064943296208838819984, −8.484378516264037348432768644731, −7.80604097105995035548927458280, −6.31095464701282956399213403489, −5.02270909955010252979871011808, −4.83325807295692173920753371230, −3.87857896447073635315498801478, −2.46066577083743670762932046404, −1.40176069244099492316005153129, −0.18890245370169336375534339194, 2.31291787234767368585972327546, 3.32685850671220338687995147555, 4.30530776517581053640086842033, 5.43051921842333032957900690151, 6.37740478575496866121812422003, 6.90655042028833515328307090098, 7.51003541078299752915522088937, 8.341281833193170729711527303137, 9.398275790711519551742976708899, 10.29406552428233469191472194390

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