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Properties

Label 2-702-117.25-c1-0-1
Degree $2$
Conductor $702$
Sign $0.0514 - 0.998i$
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 + (−0.515 + 0.297i)5-s + (−1.45 − 0.838i)7-s − 0.999i·8-s + 0.594·10-s + (0.416 + 0.240i)11-s + (−2.27 + 2.79i)13-s + (0.838 + 1.45i)14-s + (−0.5 + 0.866i)16-s + 2.09·17-s + 0.480i·19-s + (−0.515 − 0.297i)20-s + (−0.240 − 0.416i)22-s + (1.83 + 3.17i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.230 + 0.132i)5-s + (−0.548 − 0.316i)7-s − 0.353i·8-s + 0.188·10-s + (0.125 + 0.0724i)11-s + (−0.630 + 0.776i)13-s + (0.224 + 0.388i)14-s + (−0.125 + 0.216i)16-s + 0.507·17-s + 0.110i·19-s + (−0.115 − 0.0664i)20-s + (−0.0512 − 0.0887i)22-s + (0.382 + 0.662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0514 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0514 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.0514 - 0.998i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.0514 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449111 + 0.426561i\)
\(L(\frac12)\) \(\approx\) \(0.449111 + 0.426561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 - 2.79i)T \)
good5 \( 1 + (0.515 - 0.297i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.45 + 0.838i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.416 - 0.240i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
19 \( 1 - 0.480iT - 19T^{2} \)
23 \( 1 + (-1.83 - 3.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.993 - 0.573i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.65iT - 37T^{2} \)
41 \( 1 + (8.58 - 4.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.40 - 3.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.08T + 53T^{2} \)
59 \( 1 + (8.13 - 4.69i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.90 - 6.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.4 + 7.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.51iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 + (-1.02 + 1.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.57 - 5.53i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.48iT - 89T^{2} \)
97 \( 1 + (-8.41 - 4.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.53784950722921844131939266478, −9.717134046591096311794343673862, −9.191975476734661313342310047898, −8.054909145780345266753168008836, −7.24173409863113817090023553298, −6.52788358391031013445937285714, −5.18963101357437203783283513701, −3.91726646989132241066823472582, −2.99159139461278388439131841402, −1.51188304108553934555700350021, 0.39981135895463686272325649640, 2.28471901108494994653993376125, 3.54572538652058235297189148833, 4.96302354597972697708304777223, 5.87227407244111366386731664025, 6.81349916462597342258591268517, 7.71263894163328212274991642091, 8.490814292974824385504276073409, 9.356124067089974640907835900465, 10.12415685503754235091124806959

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