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Properties

Label 2-702-117.11-c1-0-11
Degree $2$
Conductor $702$
Sign $-0.760 + 0.649i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.35 − 0.361i)5-s + (0.977 + 0.261i)7-s + (0.707 + 0.707i)8-s + (1.21 − 0.699i)10-s + (−4.11 − 4.11i)11-s + (3.22 + 1.61i)13-s + (−0.876 + 0.505i)14-s − 1.00·16-s + (−1.67 + 2.89i)17-s + (−7.07 + 1.89i)19-s + (−0.361 + 1.35i)20-s + 5.81·22-s + (−0.290 + 0.502i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.603 − 0.161i)5-s + (0.369 + 0.0989i)7-s + (0.250 + 0.250i)8-s + (0.382 − 0.221i)10-s + (−1.23 − 1.23i)11-s + (0.894 + 0.446i)13-s + (−0.234 + 0.135i)14-s − 0.250·16-s + (−0.405 + 0.702i)17-s + (−1.62 + 0.434i)19-s + (−0.0809 + 0.301i)20-s + 1.23·22-s + (−0.0605 + 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0586624 - 0.158916i\)
\(L(\frac12)\) \(\approx\) \(0.0586624 - 0.158916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
13 \( 1 + (-3.22 - 1.61i)T \)
good5 \( 1 + (1.35 + 0.361i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.977 - 0.261i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.11 + 4.11i)T + 11iT^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.07 - 1.89i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.290 - 0.502i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.03iT - 29T^{2} \)
31 \( 1 + (-2.28 + 8.52i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.82 + 2.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.423 - 1.58i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.25 - 4.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.45 + 0.388i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 2.77iT - 53T^{2} \)
59 \( 1 + (8.47 + 8.47i)T + 59iT^{2} \)
61 \( 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.09 + 0.293i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.71 + 10.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.788 + 0.788i)T - 73iT^{2} \)
79 \( 1 + (0.827 - 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.21 - 15.7i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.783 + 2.92i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.18 + 11.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.19953803739303201928875225786, −8.911227181115109832140725616140, −8.210302481758641788693051073930, −7.934141734301669158008013744009, −6.46169061293610271800679995921, −5.87158543716321264539943007075, −4.65740978246793990361391535271, −3.59713040623507165239471185989, −1.96268524714759776524458377385, −0.099267946391601268014855845269, 1.86458940744360179304135161802, 3.03404073686163148555278710306, 4.27664881366779780372709009436, 5.12356519351037258448936098399, 6.65690133120096862664653200765, 7.47189840401343502719368624132, 8.271899416912659131246575785049, 8.954648020580573897424118479614, 10.27688699098232459096634801898, 10.56804203614466154853420807522

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