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Properties

Label 2-1397-1397.1142-c0-0-0
Degree $2$
Conductor $1397$
Sign $-0.884 + 0.466i$
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  + (−0.574 + 1.76i)2-s + (−1.98 − 1.44i)4-s + (2.19 − 1.59i)8-s + (0.309 − 0.951i)9-s + (−0.992 + 0.125i)11-s + (−0.613 + 1.88i)13-s + (0.798 + 2.45i)16-s + (0.331 + 1.01i)17-s + (1.50 + 1.09i)18-s + (−0.101 + 0.0738i)19-s + (0.348 − 1.82i)22-s + (−0.809 + 0.587i)25-s + (−2.98 − 2.16i)26-s + (−0.574 + 1.76i)31-s − 2.09·32-s + ⋯
L(s)  = 1  + (−0.574 + 1.76i)2-s + (−1.98 − 1.44i)4-s + (2.19 − 1.59i)8-s + (0.309 − 0.951i)9-s + (−0.992 + 0.125i)11-s + (−0.613 + 1.88i)13-s + (0.798 + 2.45i)16-s + (0.331 + 1.01i)17-s + (1.50 + 1.09i)18-s + (−0.101 + 0.0738i)19-s + (0.348 − 1.82i)22-s + (−0.809 + 0.587i)25-s + (−2.98 − 2.16i)26-s + (−0.574 + 1.76i)31-s − 2.09·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $-0.884 + 0.466i$
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: \((2,\ 1397,\ (\ :0),\ -0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4633715478\)
\(L(\frac12)\) \(\approx\) \(0.4633715478\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.992 - 0.125i)T \)
127 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.598 - 1.84i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.303 - 0.220i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \)
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

−9.946993647120426093905352608813, −9.009316674737866372211519690952, −8.715914210453680820370540299512, −7.54651419953781860001066250919, −7.07657904636046453713852122062, −6.35710928062645260321223129579, −5.51754639149520742294349773994, −4.65421879027008546538617577274, −3.72664546446922990376318786464, −1.67680548237131592296696975258, 0.46423514148740273336958818866, 2.16689610767286010915389261365, 2.74522488822718029764790719083, 3.75585363849431664398457674740, 4.95521288951420948144944338578, 5.47782160963230288866100598570, 7.46312398542500949659306729961, 7.894975679744443705134708639232, 8.606435750969307557857632439498, 9.824627320787318894682633011140

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