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Properties

Label 2-1368-152.75-c1-0-59
Degree $2$
Conductor $1368$
Sign $0.727 + 0.685i$
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  + (1.11 − 0.866i)2-s + (0.500 − 1.93i)4-s + 3.87i·7-s + (−1.11 − 2.59i)8-s + 4·11-s − 2.23·13-s + (3.35 + 4.33i)14-s + (−3.5 − 1.93i)16-s + 17-s + (4 − 1.73i)19-s + (4.47 − 3.46i)22-s − 3.87i·23-s + 5·25-s + (−2.50 + 1.93i)26-s + (7.50 + 1.93i)28-s + 2.23·29-s + ⋯
L(s)  = 1  + (0.790 − 0.612i)2-s + (0.250 − 0.968i)4-s + 1.46i·7-s + (−0.395 − 0.918i)8-s + 1.20·11-s − 0.620·13-s + (0.896 + 1.15i)14-s + (−0.875 − 0.484i)16-s + 0.242·17-s + (0.917 − 0.397i)19-s + (0.953 − 0.738i)22-s − 0.807i·23-s + 25-s + (−0.490 + 0.379i)26-s + (1.41 + 0.365i)28-s + 0.415·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.727 + 0.685i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.727 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.821677665\)
\(L(\frac12)\) \(\approx\) \(2.821677665\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.866i)T \)
3 \( 1 \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 3.87iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 2.23T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 + 3.87iT - 23T^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 7.74iT - 47T^{2} \)
53 \( 1 + 6.70T + 53T^{2} \)
59 \( 1 + 8.66iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.447062214143206220929283722296, −9.049987309060003823813908521160, −7.930708567882617407854597251270, −6.57349577464226410385854756524, −6.20848033753724115396954995102, −5.06978267834444811448463449313, −4.53010677789162545793137801676, −3.13834579058895606882447116886, −2.53032812729091655675763953936, −1.19459296616006443604225780923, 1.19066901621393516012716199949, 2.93511767515816796977936814705, 3.87593267856767494894255118184, 4.50023377161755899920378762654, 5.49015967690976874592856078266, 6.55069464738525461708768651802, 7.13094243391789382749046352794, 7.74396816610723517797134606431, 8.714608896157915284953518240914, 9.710629565779120571118804823427

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