Properties

Label 2-1397-1397.1269-c0-0-2
Degree $2$
Conductor $1397$
Sign $-0.962 + 0.271i$
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  + (−1.17 − 0.856i)2-s + (0.347 + 1.07i)4-s + (0.0565 − 0.174i)8-s + (−0.809 − 0.587i)9-s + (0.968 − 0.248i)11-s + (−1.56 − 1.13i)13-s + (0.694 − 0.504i)16-s + (0.688 − 0.500i)17-s + (0.450 + 1.38i)18-s + (−0.613 + 1.88i)19-s + (−1.35 − 0.536i)22-s + (0.309 − 0.951i)25-s + (0.872 + 2.68i)26-s + (−1.17 − 0.856i)31-s − 1.43·32-s + ⋯
L(s)  = 1  + (−1.17 − 0.856i)2-s + (0.347 + 1.07i)4-s + (0.0565 − 0.174i)8-s + (−0.809 − 0.587i)9-s + (0.968 − 0.248i)11-s + (−1.56 − 1.13i)13-s + (0.694 − 0.504i)16-s + (0.688 − 0.500i)17-s + (0.450 + 1.38i)18-s + (−0.613 + 1.88i)19-s + (−1.35 − 0.536i)22-s + (0.309 − 0.951i)25-s + (0.872 + 2.68i)26-s + (−1.17 − 0.856i)31-s − 1.43·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $-0.962 + 0.271i$
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: \((2,\ 1397,\ (\ :0),\ -0.962 + 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3670803724\)
\(L(\frac12)\) \(\approx\) \(0.3670803724\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.968 + 0.248i)T \)
127 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
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

−9.422393597836410703732946839282, −8.838369929488510691791026545547, −7.982204963325342831689811606066, −7.38065088753978877439874249586, −6.03345759141513878113036344287, −5.38636893992701252105030436601, −3.84527779932840792819354438938, −2.96718715042529943969156398945, −1.94434819629363657821765438967, −0.44154511823781459240546527747, 1.61962578846122711736956143991, 2.96163823145744503083615784555, 4.43178640788131736069864840627, 5.29538259305579538121867306897, 6.46891249628964642258367897935, 6.99606520339165999644495780050, 7.68471864372482847085307775050, 8.607272356195134689615677160343, 9.279687934543678936621696708668, 9.620680060696850981060614007193

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