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Properties

Label 2-1397-1397.1142-c0-0-4
Degree $2$
Conductor $1397$
Sign $-0.717 - 0.696i$
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  + (0.598 − 1.84i)2-s + (−2.22 − 1.61i)4-s + (−2.74 + 1.99i)8-s + (0.309 − 0.951i)9-s + (−0.425 − 0.904i)11-s + (−0.263 + 0.809i)13-s + (1.18 + 3.63i)16-s + (−0.574 − 1.76i)17-s + (−1.56 − 1.13i)18-s + (−0.866 + 0.629i)19-s + (−1.92 + 0.242i)22-s + (−0.809 + 0.587i)25-s + (1.33 + 0.969i)26-s + (0.598 − 1.84i)31-s + 4.01·32-s + ⋯
L(s)  = 1  + (0.598 − 1.84i)2-s + (−2.22 − 1.61i)4-s + (−2.74 + 1.99i)8-s + (0.309 − 0.951i)9-s + (−0.425 − 0.904i)11-s + (−0.263 + 0.809i)13-s + (1.18 + 3.63i)16-s + (−0.574 − 1.76i)17-s + (−1.56 − 1.13i)18-s + (−0.866 + 0.629i)19-s + (−1.92 + 0.242i)22-s + (−0.809 + 0.587i)25-s + (1.33 + 0.969i)26-s + (0.598 − 1.84i)31-s + 4.01·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(0.697193\)
Root analytic conductor: \(0.834981\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1397} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1397,\ (\ :0),\ -0.717 - 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004339862\)
\(L(\frac12)\) \(\approx\) \(1.004339862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.425 + 0.904i)T \)
127 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.491741575776288402410163674707, −9.029038662017943925789303200949, −7.895465255984451554859391961455, −6.45457625419869206813673957239, −5.68290190039428038847602288687, −4.57608011519359539493901548178, −3.99472798036209074277804230307, −2.97690739319855480567415938743, −2.14438922641477184279723965662, −0.68675180098394359613356607627, 2.44014419866001410825607614683, 3.98031067704439479291977772160, 4.58674975742033184314551782340, 5.36539551001715142625442107357, 6.21504972916340941622588828000, 6.95459409704904693641062847025, 7.82387963357671942282724199818, 8.192422050214640744805831167260, 9.057210106628456015491947292689, 10.13602436644645356765515440327

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