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Properties

Label 2-1397-1397.1269-c0-0-3
Degree $2$
Conductor $1397$
Sign $0.938 + 0.345i$
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  + (0.688 + 0.500i)2-s + (−0.0849 − 0.261i)4-s + (0.335 − 1.03i)8-s + (−0.809 − 0.587i)9-s + (−0.929 − 0.368i)11-s + (1.50 + 1.09i)13-s + (0.525 − 0.381i)16-s + (1.60 − 1.16i)17-s + (−0.263 − 0.809i)18-s + (−0.115 + 0.356i)19-s + (−0.456 − 0.718i)22-s + (0.309 − 0.951i)25-s + (0.489 + 1.50i)26-s + (0.688 + 0.500i)31-s − 0.532·32-s + ⋯
L(s)  = 1  + (0.688 + 0.500i)2-s + (−0.0849 − 0.261i)4-s + (0.335 − 1.03i)8-s + (−0.809 − 0.587i)9-s + (−0.929 − 0.368i)11-s + (1.50 + 1.09i)13-s + (0.525 − 0.381i)16-s + (1.60 − 1.16i)17-s + (−0.263 − 0.809i)18-s + (−0.115 + 0.356i)19-s + (−0.456 − 0.718i)22-s + (0.309 − 0.951i)25-s + (0.489 + 1.50i)26-s + (0.688 + 0.500i)31-s − 0.532·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(0.697193\)
Root analytic conductor: \(0.834981\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1397} (1269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1397,\ (\ :0),\ 0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457521041\)
\(L(\frac12)\) \(\approx\) \(1.457521041\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.929 + 0.368i)T \)
127 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
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

−9.637909365712810385121678827661, −8.914858890836413397976323807235, −8.049757326783003026960612434267, −7.08437959092690380313274064076, −6.11458177451151387589400691531, −5.75805371383987858428307607469, −4.79525804527664874738681877138, −3.78549466394834357687534345979, −2.91249933296821215093927737906, −1.10555176126170837305144471810, 1.74000522036254381379053028419, 3.16343365449364781285224844104, 3.38496100239401896441260189701, 4.83794390122589401793299561526, 5.42385121431677561689830178583, 6.22007166728769602348500027721, 7.78165116020793447402154976974, 8.084687169350964644772530058833, 8.751804833183774461677328228699, 10.21806702948928637793082824232

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