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Properties

Label 2-702-117.25-c1-0-11
Degree $2$
Conductor $702$
Sign $0.915 - 0.401i$
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 + (3.32 − 1.91i)5-s + (2.91 + 1.68i)7-s + 0.999i·8-s + 3.83·10-s + (−1.72 − 0.995i)11-s + (−0.985 + 3.46i)13-s + (1.68 + 2.91i)14-s + (−0.5 + 0.866i)16-s − 2.60·17-s − 1.99i·19-s + (3.32 + 1.91i)20-s + (−0.995 − 1.72i)22-s + (−2.13 − 3.70i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.48 − 0.857i)5-s + (1.10 + 0.636i)7-s + 0.353i·8-s + 1.21·10-s + (−0.520 − 0.300i)11-s + (−0.273 + 0.961i)13-s + (0.450 + 0.779i)14-s + (−0.125 + 0.216i)16-s − 0.630·17-s − 0.456i·19-s + (0.742 + 0.428i)20-s + (−0.212 − 0.367i)22-s + (−0.445 − 0.771i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.915 - 0.401i$
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: \((2,\ 702,\ (\ :1/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78495 + 0.583957i\)
\(L(\frac12)\) \(\approx\) \(2.78495 + 0.583957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.985 - 3.46i)T \)
good5 \( 1 + (-3.32 + 1.91i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.91 - 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.72 + 0.995i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 + 1.99iT - 19T^{2} \)
23 \( 1 + (2.13 + 3.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.57 - 2.64i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.08iT - 37T^{2} \)
41 \( 1 + (7.13 - 4.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.13 - 8.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.22 - 2.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.16T + 53T^{2} \)
59 \( 1 + (6.33 - 3.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.63 + 9.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.90 - 2.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + 8.41iT - 73T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.03 + 1.17i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.28iT - 89T^{2} \)
97 \( 1 + (-8.92 - 5.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.49329106965358177112813466638, −9.493633959627053426505221472692, −8.693590378960872856400238660441, −8.083848956415328018789843838833, −6.64420699371693760825657929269, −5.93230452095905175612769331238, −4.95698165547675454332579624652, −4.58042017878581936248557858870, −2.55751984246238684739736091141, −1.73743304012477871261292717326, 1.64364451153649180845611022153, 2.51109349615948189978294633726, 3.76793712348680592027879039510, 5.18820499660372291903209677498, 5.57144221118900571024875539723, 6.82626582693723527652113044004, 7.51745445272020351177668543958, 8.773250495265690415106729057861, 9.994033435485890832241983364867, 10.47046160517532647180369287414

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