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Properties

Label 2-702-117.103-c1-0-0
Degree $2$
Conductor $702$
Sign $-0.631 - 0.775i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.419 − 0.242i)5-s + (−4.37 + 2.52i)7-s − 0.999i·8-s − 0.484·10-s + (−2.78 + 1.60i)11-s + (−2.69 + 2.39i)13-s + (−2.52 + 4.37i)14-s + (−0.5 − 0.866i)16-s − 4.20·17-s + 3.21i·19-s + (−0.419 + 0.242i)20-s + (−1.60 + 2.78i)22-s + (3.13 − 5.43i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.187 − 0.108i)5-s + (−1.65 + 0.955i)7-s − 0.353i·8-s − 0.153·10-s + (−0.838 + 0.484i)11-s + (−0.748 + 0.663i)13-s + (−0.675 + 1.16i)14-s + (−0.125 − 0.216i)16-s − 1.01·17-s + 0.736i·19-s + (−0.0937 + 0.0541i)20-s + (−0.342 + 0.592i)22-s + (0.653 − 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204226 + 0.429484i\)
\(L(\frac12)\) \(\approx\) \(0.204226 + 0.429484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
13 \( 1 + (2.69 - 2.39i)T \)
good5 \( 1 + (0.419 + 0.242i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (4.37 - 2.52i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.78 - 1.60i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 - 3.21iT - 19T^{2} \)
23 \( 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 3.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.61 - 3.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.08iT - 37T^{2} \)
41 \( 1 + (9.57 + 5.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.57 - 2.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 + (-3.13 - 1.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.936 + 0.540i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + 0.325iT - 73T^{2} \)
79 \( 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 - 2.93i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.42iT - 89T^{2} \)
97 \( 1 + (11.3 - 6.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.67278379429178454209509857120, −9.960534285918928967329891464562, −9.234592667267716340702479494409, −8.284045664379738113592657519627, −6.83773440265483659388760886241, −6.40255903333052342627001268063, −5.22828561398951896532092644053, −4.33112163263410638532679615608, −3.01015524717142024409517368296, −2.30155635757963874050790962650, 0.18386750712169190327341470860, 2.77045732956011498307999739790, 3.45884369877892890602650899441, 4.61157382350703034750714461498, 5.67646364497401602550835552117, 6.68108029837614979575747574003, 7.25252872878618271059258547667, 8.170490247908312765425973785015, 9.434981011887080932736844173018, 10.12104699506930228942792779353

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