Properties

Label 2-1397-1397.126-c0-0-1
Degree $2$
Conductor $1397$
Sign $0.441 + 0.897i$
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  + (−0.393 − 1.21i)2-s + (−0.505 + 0.367i)4-s + (−0.386 − 0.280i)8-s + (0.309 + 0.951i)9-s + (0.728 + 0.684i)11-s + (0.450 + 1.38i)13-s + (−0.381 + 1.17i)16-s + (0.598 − 1.84i)17-s + (1.03 − 0.749i)18-s + (1.50 + 1.09i)19-s + (0.542 − 1.15i)22-s + (−0.809 − 0.587i)25-s + (1.50 − 1.09i)26-s + (−0.393 − 1.21i)31-s + 1.09·32-s + ⋯
L(s)  = 1  + (−0.393 − 1.21i)2-s + (−0.505 + 0.367i)4-s + (−0.386 − 0.280i)8-s + (0.309 + 0.951i)9-s + (0.728 + 0.684i)11-s + (0.450 + 1.38i)13-s + (−0.381 + 1.17i)16-s + (0.598 − 1.84i)17-s + (1.03 − 0.749i)18-s + (1.50 + 1.09i)19-s + (0.542 − 1.15i)22-s + (−0.809 − 0.587i)25-s + (1.50 − 1.09i)26-s + (−0.393 − 1.21i)31-s + 1.09·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1397 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1397\)    =    \(11 \cdot 127\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(0.697193\)
Root analytic conductor: \(0.834981\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1397} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1397,\ (\ :0),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9737140308\)
\(L(\frac12)\) \(\approx\) \(0.9737140308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.728 - 0.684i)T \)
127 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.559733063768340879587387094701, −9.415896282577871282491100082907, −8.120637416713614429039315296611, −7.29058704041115480504648973073, −6.45468777140066049207161873443, −5.25698986479370966944872758536, −4.26556692667367364916068804390, −3.36575331886818632033332491287, −2.17209863143811233063108689247, −1.39648918093132898370405970812, 1.13591963817489226489464160794, 3.15476305488597045868825373691, 3.80129584952919897523217020461, 5.49356004688388512651146325440, 5.79023029647843331902353969573, 6.72442944955477896815552118566, 7.44913962992664639928488724679, 8.215939409197256186263827943979, 8.952549563536101162837032436514, 9.515263426853508289464948378866

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