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Properties

Label 2-702-117.25-c1-0-8
Degree $2$
Conductor $702$
Sign $0.705 - 0.709i$
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 + (3.56 − 0.567i)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 + (0.987 − 0.157i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.705 - 0.709i$
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: \((2,\ 702,\ (\ :1/2),\ 0.705 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22942 + 0.926973i\)
\(L(\frac12)\) \(\approx\) \(2.22942 + 0.926973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.56 + 0.567i)T \)
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

−10.78315197199849027285797701497, −9.592802863654586792566496833266, −8.701383467181831041511665008553, −7.893027926862709056775711823610, −7.00326527939483287440494406414, −5.81725365099798842751864612680, −5.34165087034139344416376017067, −4.12762690748484553119969385242, −3.08103187149462782661534812945, −1.60151343336287990313854079832, 1.28606684486683051752915762382, 2.62710185549187051602414742813, 3.85236720584731733517114644776, 4.72640719593746381417411072360, 5.81540534142483901229386943669, 6.58542408322138423306129744978, 7.73154410452094398895859850978, 8.579780653122690991659119920374, 9.717066732556802214401857735673, 10.49410209052021618177041972864

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