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Properties

Label 2-1397-1397.1015-c0-0-3
Degree $2$
Conductor $1397$
Sign $0.618 + 0.785i$
Analytic cond. $0.697193$
Root an. cond. $0.834981$
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  + (−1.41 + 1.03i)2-s + (0.640 − 1.97i)4-s + (0.580 + 1.78i)8-s + (−0.809 + 0.587i)9-s + (−0.637 − 0.770i)11-s + (1.03 − 0.749i)13-s + (−0.987 − 0.717i)16-s + (−1.17 − 0.856i)17-s + (0.541 − 1.66i)18-s + (−0.263 − 0.809i)19-s + (1.69 + 0.435i)22-s + (0.309 + 0.951i)25-s + (−0.690 + 2.12i)26-s + (−1.41 + 1.03i)31-s + 0.260·32-s + ⋯
L(s)  = 1  + (−1.41 + 1.03i)2-s + (0.640 − 1.97i)4-s + (0.580 + 1.78i)8-s + (−0.809 + 0.587i)9-s + (−0.637 − 0.770i)11-s + (1.03 − 0.749i)13-s + (−0.987 − 0.717i)16-s + (−1.17 − 0.856i)17-s + (0.541 − 1.66i)18-s + (−0.263 − 0.809i)19-s + (1.69 + 0.435i)22-s + (0.309 + 0.951i)25-s + (−0.690 + 2.12i)26-s + (−1.41 + 1.03i)31-s + 0.260·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(0.697193\)
Root analytic conductor: \(0.834981\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1397} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1397,\ (\ :0),\ 0.618 + 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3032490644\)
\(L(\frac12)\) \(\approx\) \(0.3032490644\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.637 + 0.770i)T \)
127 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.303 - 0.220i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.171924039404540997607139542561, −8.777058776918108897191334760868, −8.215740124974206381882904897677, −7.30770843713503556976455564491, −6.67571251728375063137400879157, −5.55238288557134928797650094547, −5.25961424336156844967600119163, −3.40913804171558437028679654749, −2.11910562528975565077314174877, −0.39248191546864405841234853337, 1.52033901377789150030990754774, 2.44650713843641582814010327499, 3.51284477479832423689044051210, 4.47934445092632845804671959182, 6.07531893067484011221214710293, 6.73763885416864500792089090997, 8.118547220779990775490554416179, 8.262003125600870590563365384515, 9.259667684857055144109525049676, 9.744352124107770712570276050517

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