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Properties

Label 2-702-117.32-c1-0-1
Degree $2$
Conductor $702$
Sign $-0.949 + 0.315i$
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 + (−0.653 + 0.175i)5-s + (−3.90 + 1.04i)7-s + (−0.707 + 0.707i)8-s + (−0.585 − 0.338i)10-s + (0.502 − 0.502i)11-s + (−2.29 + 2.77i)13-s + (−3.49 − 2.01i)14-s − 1.00·16-s + (−3.24 − 5.61i)17-s + (−0.253 − 0.0679i)19-s + (−0.175 − 0.653i)20-s + 0.710·22-s + (−0.860 − 1.49i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.292 + 0.0782i)5-s + (−1.47 + 0.394i)7-s + (−0.250 + 0.250i)8-s + (−0.185 − 0.106i)10-s + (0.151 − 0.151i)11-s + (−0.637 + 0.770i)13-s + (−0.934 − 0.539i)14-s − 0.250·16-s + (−0.786 − 1.36i)17-s + (−0.0581 − 0.0155i)19-s + (−0.0391 − 0.146i)20-s + 0.151·22-s + (−0.179 − 0.310i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.949 + 0.315i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.949 + 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0716758 - 0.443481i\)
\(L(\frac12)\) \(\approx\) \(0.0716758 - 0.443481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 - 2.77i)T \)
good5 \( 1 + (0.653 - 0.175i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.90 - 1.04i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.502 + 0.502i)T - 11iT^{2} \)
17 \( 1 + (3.24 + 5.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.253 + 0.0679i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.860 + 1.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.28iT - 29T^{2} \)
31 \( 1 + (-1.04 - 3.89i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.96 - 2.13i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.39 - 8.94i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.67 + 3.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.07 - 2.43i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 6.34iT - 53T^{2} \)
59 \( 1 + (-3.52 + 3.52i)T - 59iT^{2} \)
61 \( 1 + (1.64 - 2.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.0 - 2.70i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.09 + 7.82i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.40 + 1.40i)T + 73iT^{2} \)
79 \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.04 - 3.90i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.64 - 9.85i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.520 - 1.94i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.06981799055288198368250113273, −9.769735452453013455138216114103, −9.282597885666945050344971699306, −8.295582498845712410244352904994, −6.96412470098666158788830334339, −6.75167240978972888074488785217, −5.59920681469658752994361731358, −4.55888185989330658746537609765, −3.46948813339658232397200899562, −2.50314576324753801814143923249, 0.18331535643574576747243577127, 2.17324723788749272587626963167, 3.47706439820134354859951694677, 4.07362657029656264677000338667, 5.42190270048500667342224396990, 6.34700166799804790716485517585, 7.14364343884630910501944905512, 8.306596455324476678014167549676, 9.356121850847556945546022897042, 10.18926817635784965221583357042

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