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Properties

Label 2-1368-152.75-c1-0-80
Degree $2$
Conductor $1368$
Sign $0.458 + 0.888i$
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  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s − 3.46i·7-s − 2.82·8-s + 3.16·11-s − 4.47·13-s + (4.24 − 2.44i)14-s + (−2.00 − 3.46i)16-s − 6.32·17-s + (−2 − 3.87i)19-s + (2.23 + 3.87i)22-s − 5.47i·23-s + 5·25-s + (−3.16 − 5.47i)26-s + (5.99 + 3.46i)28-s + 1.41·29-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.30i·7-s − 0.999·8-s + 0.953·11-s − 1.24·13-s + (1.13 − 0.654i)14-s + (−0.500 − 0.866i)16-s − 1.53·17-s + (−0.458 − 0.888i)19-s + (0.476 + 0.825i)22-s − 1.14i·23-s + 25-s + (−0.620 − 1.07i)26-s + (1.13 + 0.654i)28-s + 0.262·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.458 + 0.888i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028656909\)
\(L(\frac12)\) \(\approx\) \(1.028656909\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
19 \( 1 + (2 + 3.87i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 6.32T + 17T^{2} \)
23 \( 1 + 5.47iT - 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 5.47iT - 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 4.89iT - 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + 17.1iT - 89T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−9.084684327136900152159594613337, −8.723866923983568627481418772116, −7.46511639437992713802876708388, −6.90408974546402786167154570600, −6.51054670218153364787813263388, −5.05524401416802407371757564597, −4.45620135782952356698681839059, −3.71483568082702740303754892285, −2.37578242542297392651074979599, −0.33605209465174742598442123407, 1.72146450280527091679784151152, 2.49502210378172303780187543777, 3.59906498643372312398371076186, 4.61435453581979584980780912071, 5.40293137256400105181750905176, 6.21952216041146914158712312692, 7.11475622131473451257136178790, 8.508990198784454971286445740911, 9.148709622855554642300584860684, 9.612549164849985230605535607009

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