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Properties

Label 2-702-117.110-c1-0-9
Degree $2$
Conductor $702$
Sign $-0.301 + 0.953i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.266 + 0.994i)5-s + (−0.248 − 0.248i)7-s + (0.707 − 0.707i)8-s + (−0.891 − 0.514i)10-s + (−3.50 − 0.940i)11-s + (−3.13 − 1.78i)13-s + (0.304 − 0.175i)14-s + (0.500 + 0.866i)16-s + (−1.67 − 2.89i)17-s + (−0.969 − 0.259i)19-s + (0.727 − 0.727i)20-s + (1.81 − 3.14i)22-s − 1.47·23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.119 + 0.444i)5-s + (−0.0938 − 0.0938i)7-s + (0.249 − 0.249i)8-s + (−0.281 − 0.162i)10-s + (−1.05 − 0.283i)11-s + (−0.869 − 0.494i)13-s + (0.0812 − 0.0469i)14-s + (0.125 + 0.216i)16-s + (−0.405 − 0.702i)17-s + (−0.222 − 0.0596i)19-s + (0.162 − 0.162i)20-s + (0.387 − 0.670i)22-s − 0.306·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.301 + 0.953i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.301 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140133 - 0.191225i\)
\(L(\frac12)\) \(\approx\) \(0.140133 - 0.191225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (0.266 - 0.994i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.248 + 0.248i)T + 7iT^{2} \)
11 \( 1 + (3.50 + 0.940i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.969 + 0.259i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 1.47T + 23T^{2} \)
29 \( 1 + (5.04 - 2.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.54 + 1.48i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.96 + 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.60 + 6.60i)T + 41iT^{2} \)
43 \( 1 + 2.73iT - 43T^{2} \)
47 \( 1 + (2.19 + 8.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 2.67iT - 53T^{2} \)
59 \( 1 + (1.25 + 4.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 + (1.53 - 5.74i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.47 - 2.47i)T + 73iT^{2} \)
79 \( 1 + (1.76 - 3.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.34 + 2.23i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.68 + 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.4 - 10.4i)T - 97iT^{2} \)
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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

−10.19374128797538107682812569670, −9.249950854101141124274498980530, −8.351950023020488252500253761659, −7.38636959544674592324160459267, −6.95255564718573697102058509883, −5.61443484484880455832163185288, −5.01346414185279192949339927234, −3.60931749951168079021061612019, −2.38553501105456528181637725893, −0.12385430087910752478422368989, 1.81160707924032527594810137260, 2.90714174505915184046107341575, 4.28526827767735360464413740969, 5.00024409848269580917885921294, 6.20958108621077894851641073979, 7.47856556344618967991560176222, 8.188506881680921365552948566254, 9.150780230730611060763595746377, 9.859339222048434644432065657332, 10.70205637300758840185115475116

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