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Properties

Label 2-1379-1379.1000-c0-0-0
Degree $2$
Conductor $1379$
Sign $-0.792 + 0.609i$
Analytic cond. $0.688210$
Root an. cond. $0.829584$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.750 + 1.85i)2-s + (−2.15 + 2.08i)4-s + (0.926 + 0.375i)7-s + (−3.64 − 1.61i)8-s + (−0.518 + 0.855i)9-s + (−0.856 + 0.138i)11-s + 1.99i·14-s + (0.159 − 4.97i)16-s + (−1.97 − 0.319i)18-s + (−0.899 − 1.48i)22-s + (1.03 + 1.47i)23-s + (−0.801 − 0.598i)25-s + (−2.77 + 1.12i)28-s + (−0.161 + 0.545i)29-s + (5.59 − 2.05i)32-s + ⋯
L(s)  = 1  + (0.750 + 1.85i)2-s + (−2.15 + 2.08i)4-s + (0.926 + 0.375i)7-s + (−3.64 − 1.61i)8-s + (−0.518 + 0.855i)9-s + (−0.856 + 0.138i)11-s + 1.99i·14-s + (0.159 − 4.97i)16-s + (−1.97 − 0.319i)18-s + (−0.899 − 1.48i)22-s + (1.03 + 1.47i)23-s + (−0.801 − 0.598i)25-s + (−2.77 + 1.12i)28-s + (−0.161 + 0.545i)29-s + (5.59 − 2.05i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1379\)    =    \(7 \cdot 197\)
Sign: $-0.792 + 0.609i$
Analytic conductor: \(0.688210\)
Root analytic conductor: \(0.829584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1379} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1379,\ (\ :0),\ -0.792 + 0.609i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199093741\)
\(L(\frac12)\) \(\approx\) \(1.199093741\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.926 - 0.375i)T \)
197 \( 1 + (-0.404 + 0.914i)T \)
good2 \( 1 + (-0.750 - 1.85i)T + (-0.718 + 0.695i)T^{2} \)
3 \( 1 + (0.518 - 0.855i)T^{2} \)
5 \( 1 + (0.801 + 0.598i)T^{2} \)
11 \( 1 + (0.856 - 0.138i)T + (0.949 - 0.315i)T^{2} \)
13 \( 1 + (0.926 - 0.375i)T^{2} \)
17 \( 1 + (-0.981 - 0.191i)T^{2} \)
19 \( 1 + (-0.623 - 0.781i)T^{2} \)
23 \( 1 + (-1.03 - 1.47i)T + (-0.345 + 0.938i)T^{2} \)
29 \( 1 + (0.161 - 0.545i)T + (-0.838 - 0.545i)T^{2} \)
31 \( 1 + (-0.0960 - 0.995i)T^{2} \)
37 \( 1 + (0.00615 + 0.191i)T + (-0.997 + 0.0640i)T^{2} \)
41 \( 1 + (0.981 + 0.191i)T^{2} \)
43 \( 1 + (0.255 + 1.58i)T + (-0.949 + 0.315i)T^{2} \)
47 \( 1 + (-0.871 + 0.490i)T^{2} \)
53 \( 1 + (-1.62 - 1.05i)T + (0.404 + 0.914i)T^{2} \)
59 \( 1 + (-0.991 - 0.127i)T^{2} \)
61 \( 1 + (-0.518 - 0.855i)T^{2} \)
67 \( 1 + (-0.328 - 1.25i)T + (-0.871 + 0.490i)T^{2} \)
71 \( 1 + (-1.18 - 0.721i)T + (0.462 + 0.886i)T^{2} \)
73 \( 1 + (-0.997 + 0.0640i)T^{2} \)
79 \( 1 + (0.343 + 1.03i)T + (-0.801 + 0.598i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.0960 + 0.995i)T^{2} \)
97 \( 1 + (-0.967 + 0.253i)T^{2} \)
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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

−10.00339335923806983784425657342, −8.837484994580412520965935396910, −8.461805158265181647922867265534, −7.52681939010585063221663995773, −7.27833587064411329780677339528, −5.88243319413584629821340435356, −5.36230538898508920879955443343, −4.85362382497774369921588358577, −3.77203213125142803409594716767, −2.55055266519041570710722758300, 0.806261202310483711483787913356, 2.13366187371401782506162392128, 3.03882621547790655156221804416, 3.95445071842792541335204493524, 4.82701718471694214313314543826, 5.48427007079162774898219710963, 6.47623957899907906282797791743, 8.069421901231444641365510550803, 8.770371432982637504745721965518, 9.610076502038039335679350074719

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