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Properties

Label 2-702-117.2-c1-0-1
Degree $2$
Conductor $702$
Sign $0.346 - 0.938i$
Analytic cond. $5.60549$
Root an. cond. $2.36759$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.964 − 3.60i)5-s + (−0.861 + 3.21i)7-s + (0.707 − 0.707i)8-s + (−3.22 + 1.86i)10-s + (−4.48 + 4.48i)11-s + (−0.905 + 3.49i)13-s + (2.88 − 1.66i)14-s − 1.00·16-s + (−1.38 + 2.39i)17-s + (0.703 + 2.62i)19-s + (3.60 + 0.964i)20-s + 6.34·22-s + (−0.0915 + 0.158i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.431 − 1.61i)5-s + (−0.325 + 1.21i)7-s + (0.250 − 0.250i)8-s + (−1.02 + 0.589i)10-s + (−1.35 + 1.35i)11-s + (−0.251 + 0.967i)13-s + (0.769 − 0.444i)14-s − 0.250·16-s + (−0.335 + 0.581i)17-s + (0.161 + 0.602i)19-s + (0.805 + 0.215i)20-s + 1.35·22-s + (−0.0190 + 0.0330i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.346 - 0.938i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559476 + 0.389882i\)
\(L(\frac12)\) \(\approx\) \(0.559476 + 0.389882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
13 \( 1 + (0.905 - 3.49i)T \)
good5 \( 1 + (-0.964 + 3.60i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.861 - 3.21i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.48 - 4.48i)T - 11iT^{2} \)
17 \( 1 + (1.38 - 2.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.703 - 2.62i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0915 - 0.158i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.948iT - 29T^{2} \)
31 \( 1 + (-2.14 - 0.573i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.96 + 7.33i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.27 - 0.610i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.17 - 1.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.38 - 5.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 3.43iT - 53T^{2} \)
59 \( 1 + (3.31 - 3.31i)T - 59iT^{2} \)
61 \( 1 + (-4.13 - 7.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 - 7.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.7 - 3.16i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.16 - 2.16i)T + 73iT^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.52 - 1.74i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.643 + 0.172i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.65 + 1.51i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.30467016195746734912501266288, −9.648477261608460840866076253423, −8.992983445759592440556216572600, −8.364148099504802990390560209067, −7.38090894584164026772520615818, −5.97251187259771703167037694204, −5.09159339387421275037763961580, −4.29495161012817748177620714450, −2.47806357076584442526316094104, −1.69717522830948233249489389230, 0.40112822057771670174125492678, 2.65761602345240039215628053648, 3.36779762390802410726769632681, 5.07328160542213041507480871610, 6.08773884086456548314705798032, 6.85718142147290941338837232053, 7.56260082290741163251127159102, 8.299880218734508427556399147221, 9.685940782648683344211723994576, 10.38823633997665408010222029890

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