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Properties

Label 2-1368-152.75-c1-0-75
Degree $2$
Conductor $1368$
Sign $-0.234 + 0.972i$
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  + (−0.558 + 1.29i)2-s + (−1.37 − 1.45i)4-s + 1.71i·5-s − 0.342i·7-s + (2.65 − 0.978i)8-s + (−2.23 − 0.960i)10-s + 1.65·11-s − 4.72·13-s + (0.445 + 0.191i)14-s + (−0.209 + 3.99i)16-s − 5.61·17-s + (2.42 − 3.62i)19-s + (2.49 − 2.36i)20-s + (−0.925 + 2.15i)22-s + 3.13i·23-s + ⋯
L(s)  = 1  + (−0.394 + 0.918i)2-s + (−0.688 − 0.725i)4-s + 0.769i·5-s − 0.129i·7-s + (0.938 − 0.346i)8-s + (−0.706 − 0.303i)10-s + 0.499·11-s − 1.31·13-s + (0.118 + 0.0511i)14-s + (−0.0524 + 0.998i)16-s − 1.36·17-s + (0.556 − 0.830i)19-s + (0.557 − 0.529i)20-s + (−0.197 + 0.459i)22-s + 0.653i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.234 + 0.972i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02559874305\)
\(L(\frac12)\) \(\approx\) \(0.02559874305\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.558 - 1.29i)T \)
3 \( 1 \)
19 \( 1 + (-2.42 + 3.62i)T \)
good5 \( 1 - 1.71iT - 5T^{2} \)
7 \( 1 + 0.342iT - 7T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 4.72T + 13T^{2} \)
17 \( 1 + 5.61T + 17T^{2} \)
23 \( 1 - 3.13iT - 23T^{2} \)
29 \( 1 + 1.36T + 29T^{2} \)
31 \( 1 + 6.03T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 - 0.258iT - 41T^{2} \)
43 \( 1 + 9.29T + 43T^{2} \)
47 \( 1 - 1.66iT - 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 - 6.84iT - 59T^{2} \)
61 \( 1 + 14.1iT - 61T^{2} \)
67 \( 1 + 5.78iT - 67T^{2} \)
71 \( 1 + 3.71T + 71T^{2} \)
73 \( 1 - 9.05T + 73T^{2} \)
79 \( 1 + 6.46T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 9.18iT - 89T^{2} \)
97 \( 1 + 7.65iT - 97T^{2} \)
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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

−9.322403440887779317179974645547, −8.574559172015180911522283294689, −7.40281382383612217474584190684, −7.06732259102073107381999747348, −6.34287067869887446030409522157, −5.21717465058113398581395621358, −4.50821925847253756673600365039, −3.24549225091402926322279426678, −1.88092425046449161558803734815, −0.01178612702714513691742436445, 1.51052741018978472061128642907, 2.51273365569838727325264404045, 3.73321963807514104559919599666, 4.64865601513187399584126218171, 5.31045531620853840698810255351, 6.73768345451443201042732897942, 7.58741332300211615805399253012, 8.561328584539113333052509678570, 9.051389847929207187679862575291, 9.781830741450373144584973561497

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