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Properties

Label 2-1379-1379.734-c0-0-0
Degree $2$
Conductor $1379$
Sign $-0.994 - 0.100i$
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  + (−0.408 − 0.479i)2-s + (0.0962 − 0.595i)4-s + (−0.761 − 0.648i)7-s + (−0.863 + 0.523i)8-s + (0.672 − 0.740i)9-s + (−1.94 + 0.0624i)11-s + 0.630i·14-s + (0.0320 + 0.0106i)16-s + (−0.629 − 0.0201i)18-s + (0.826 + 0.909i)22-s + (0.436 + 0.0850i)23-s + (−0.991 − 0.127i)25-s + (−0.459 + 0.390i)28-s + (−1.87 + 0.490i)29-s + (0.371 + 0.916i)32-s + ⋯
L(s)  = 1  + (−0.408 − 0.479i)2-s + (0.0962 − 0.595i)4-s + (−0.761 − 0.648i)7-s + (−0.863 + 0.523i)8-s + (0.672 − 0.740i)9-s + (−1.94 + 0.0624i)11-s + 0.630i·14-s + (0.0320 + 0.0106i)16-s + (−0.629 − 0.0201i)18-s + (0.826 + 0.909i)22-s + (0.436 + 0.0850i)23-s + (−0.991 − 0.127i)25-s + (−0.459 + 0.390i)28-s + (−1.87 + 0.490i)29-s + (0.371 + 0.916i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1379\)    =    \(7 \cdot 197\)
Sign: $-0.994 - 0.100i$
Analytic conductor: \(0.688210\)
Root analytic conductor: \(0.829584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1379} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1379,\ (\ :0),\ -0.994 - 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4347832540\)
\(L(\frac12)\) \(\approx\) \(0.4347832540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.761 + 0.648i)T \)
197 \( 1 + (-0.518 - 0.855i)T \)
good2 \( 1 + (0.408 + 0.479i)T + (-0.159 + 0.987i)T^{2} \)
3 \( 1 + (-0.672 + 0.740i)T^{2} \)
5 \( 1 + (0.991 + 0.127i)T^{2} \)
11 \( 1 + (1.94 - 0.0624i)T + (0.997 - 0.0640i)T^{2} \)
13 \( 1 + (-0.761 + 0.648i)T^{2} \)
17 \( 1 + (-0.345 + 0.938i)T^{2} \)
19 \( 1 + (0.900 - 0.433i)T^{2} \)
23 \( 1 + (-0.436 - 0.0850i)T + (0.926 + 0.375i)T^{2} \)
29 \( 1 + (1.87 - 0.490i)T + (0.871 - 0.490i)T^{2} \)
31 \( 1 + (-0.572 + 0.820i)T^{2} \)
37 \( 1 + (-1.08 + 0.360i)T + (0.801 - 0.598i)T^{2} \)
41 \( 1 + (0.345 - 0.938i)T^{2} \)
43 \( 1 + (0.0635 + 1.98i)T + (-0.997 + 0.0640i)T^{2} \)
47 \( 1 + (-0.404 - 0.914i)T^{2} \)
53 \( 1 + (-1.46 + 0.822i)T + (0.518 - 0.855i)T^{2} \)
59 \( 1 + (-0.284 - 0.958i)T^{2} \)
61 \( 1 + (0.672 + 0.740i)T^{2} \)
67 \( 1 + (0.759 - 1.16i)T + (-0.404 - 0.914i)T^{2} \)
71 \( 1 + (1.46 + 1.32i)T + (0.0960 + 0.995i)T^{2} \)
73 \( 1 + (0.801 - 0.598i)T^{2} \)
79 \( 1 + (0.0628 + 0.979i)T + (-0.991 + 0.127i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.572 - 0.820i)T^{2} \)
97 \( 1 + (0.838 + 0.545i)T^{2} \)
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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

−9.563296304577060346156575715422, −8.856460876736885584802817591367, −7.62197042935024872864841729164, −7.07690024112354950938746108723, −5.95256032597489445887521549068, −5.33257485426178246928624438875, −4.06020792628653667513263932685, −3.03659363060670388614660983832, −1.94050037565215974406720422513, −0.37892465497635334145487615635, 2.30783435592703857527990416711, 3.03567404093206782805342082680, 4.25654881922101532093123317732, 5.42790017029911821819856226274, 6.09604591678723210863530735361, 7.27601794073377682183474843385, 7.71745213652739625562929288468, 8.374911480259134026514266561442, 9.413322842830118930374919139832, 9.927601118238905799276424216406

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