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Properties

Label 2-702-13.3-c1-0-12
Degree $2$
Conductor $702$
Sign $0.964 - 0.265i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 2.30·5-s + (1.65 − 2.86i)7-s − 0.999·8-s + (1.15 + 1.99i)10-s + (−0.348 − 0.603i)11-s + (1.80 − 3.12i)13-s + 3.30·14-s + (−0.5 − 0.866i)16-s + (1.95 − 3.38i)17-s + (−0.197 + 0.341i)19-s + (−1.15 + 1.99i)20-s + (0.348 − 0.603i)22-s + (0.802 + 1.39i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.02·5-s + (0.624 − 1.08i)7-s − 0.353·8-s + (0.364 + 0.630i)10-s + (−0.105 − 0.182i)11-s + (0.499 − 0.866i)13-s + 0.882·14-s + (−0.125 − 0.216i)16-s + (0.473 − 0.820i)17-s + (−0.0452 + 0.0783i)19-s + (−0.257 + 0.445i)20-s + (0.0743 − 0.128i)22-s + (0.167 + 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20937 + 0.298078i\)
\(L(\frac12)\) \(\approx\) \(2.20937 + 0.298078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
13 \( 1 + (-1.80 + 3.12i)T \)
good5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 + (-1.65 + 2.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.348 + 0.603i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.197 - 0.341i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.802 - 1.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.80 - 6.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.21T + 31T^{2} \)
37 \( 1 + (3.65 + 6.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.40 - 9.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.302 - 0.524i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (7.15 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.84 - 4.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 5.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.34 - 5.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 1.90T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (8.40 + 14.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.90 - 8.50i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.51704440999512226450231814508, −9.598248352773422311182738289456, −8.674739355117514559701129834657, −7.65474017058673662464117824566, −7.04978983255978034289662667015, −5.83683665687215791952549977896, −5.27786471653958429536602100601, −4.13405019361891599841001095803, −2.94295667457664057063841697905, −1.24149652574381936140579056937, 1.69428987950209548240702949921, 2.36715786900079416652273695234, 3.83411503509831759783413542754, 5.02963971157197880670249197722, 5.78118850307618146031729945867, 6.53038239759625173192531350412, 8.036264461569206541386794302055, 8.942627431177712714831083123820, 9.552934366082786627613081045250, 10.49580853129725126603180970912

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