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Properties

Label 2-1372-7.4-c1-0-7
Degree $2$
Conductor $1372$
Sign $0.5 - 0.866i$
Analytic cond. $10.9554$
Root an. cond. $3.30990$
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  + (1.02 − 1.78i)3-s + (1.41 + 2.45i)5-s + (−0.621 − 1.07i)9-s + (−3.23 + 5.61i)11-s − 1.96·13-s + 5.84·15-s + (−1.39 + 2.41i)17-s + (0.151 + 0.263i)19-s + (2.69 + 4.66i)23-s + (−1.51 + 2.63i)25-s + 3.61·27-s − 2.13·29-s + (−3.28 + 5.69i)31-s + (6.67 + 11.5i)33-s + (−3.87 − 6.70i)37-s + ⋯
L(s)  = 1  + (0.594 − 1.02i)3-s + (0.633 + 1.09i)5-s + (−0.207 − 0.358i)9-s + (−0.976 + 1.69i)11-s − 0.544·13-s + 1.50·15-s + (−0.337 + 0.584i)17-s + (0.0348 + 0.0603i)19-s + (0.561 + 0.973i)23-s + (−0.303 + 0.526i)25-s + 0.696·27-s − 0.395·29-s + (−0.590 + 1.02i)31-s + (1.16 + 2.01i)33-s + (−0.636 − 1.10i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1372\)    =    \(2^{2} \cdot 7^{3}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(10.9554\)
Root analytic conductor: \(3.30990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1372} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1372,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867256528\)
\(L(\frac12)\) \(\approx\) \(1.867256528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-1.02 + 1.78i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.41 - 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.23 - 5.61i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.96T + 13T^{2} \)
17 \( 1 + (1.39 - 2.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.151 - 0.263i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.69 - 4.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.13T + 29T^{2} \)
31 \( 1 + (3.28 - 5.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.51T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + (-5.48 - 9.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.43 + 7.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.91 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.80 - 8.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.02 + 1.77i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.91T + 71T^{2} \)
73 \( 1 + (-1.56 + 2.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.40 + 2.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.482T + 83T^{2} \)
89 \( 1 + (7.00 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−9.848928031607493792041245020584, −8.874392877320328755408316162897, −7.80661427229906190142971590502, −7.16761853985199114377683390545, −6.89748301920032014757580844979, −5.69266999257344098865495282584, −4.73576477599197874896413517196, −3.28791894118216002164384705823, −2.28392669705322812149319031547, −1.84088994615320040970018657644, 0.67168468751493980764490586035, 2.42763432434685447187090293833, 3.31815686051053838256097078596, 4.42105123463863343623896889506, 5.18564906807172398935681763328, 5.78948574841764101050568633698, 7.06355617160587078361117489440, 8.350915540988729426873429245603, 8.670174569557857871084599147904, 9.382914018624827063680317937685

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