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Properties

Label 2-702-117.59-c1-0-13
Degree $2$
Conductor $702$
Sign $-0.991 + 0.126i$
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 + (−0.891 − 3.32i)5-s + (−0.00737 − 0.0275i)7-s + (−0.707 − 0.707i)8-s + (−2.98 − 1.72i)10-s + (−3.07 − 3.07i)11-s + (−3.57 + 0.450i)13-s + (−0.0246 − 0.0142i)14-s − 1.00·16-s + (3.35 + 5.81i)17-s + (−0.521 + 1.94i)19-s + (−3.32 + 0.891i)20-s − 4.35·22-s + (−0.264 − 0.457i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.398 − 1.48i)5-s + (−0.00278 − 0.0104i)7-s + (−0.250 − 0.250i)8-s + (−0.942 − 0.544i)10-s + (−0.927 − 0.927i)11-s + (−0.992 + 0.124i)13-s + (−0.00659 − 0.00380i)14-s − 0.250·16-s + (0.813 + 1.40i)17-s + (−0.119 + 0.446i)19-s + (−0.743 + 0.199i)20-s − 0.927·22-s + (−0.0550 − 0.0953i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0780260 - 1.22934i\)
\(L(\frac12)\) \(\approx\) \(0.0780260 - 1.22934i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.450i)T \)
good5 \( 1 + (0.891 + 3.32i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.00737 + 0.0275i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.07 + 3.07i)T + 11iT^{2} \)
17 \( 1 + (-3.35 - 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.521 - 1.94i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.264 + 0.457i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.30iT - 29T^{2} \)
31 \( 1 + (-1.39 + 0.373i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.09 + 7.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.85 + 1.30i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.58 - 4.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.207 + 0.775i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 5.06iT - 53T^{2} \)
59 \( 1 + (10.1 + 10.1i)T + 59iT^{2} \)
61 \( 1 + (-0.0107 + 0.0186i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.244 - 0.912i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.32 + 2.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.53 + 6.53i)T - 73iT^{2} \)
79 \( 1 + (-1.40 - 2.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.1 - 3.80i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-10.8 + 2.91i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.455 - 0.122i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.13203298520956472650785953157, −9.227861454841249351076668876902, −8.247548410182229219320776240811, −7.74357588120400556784559339129, −6.05316172805562841553694056726, −5.36410063469955273019211458339, −4.46757362988929674172346781888, −3.54158891073335368351461784778, −2.05352364532789707762161955405, −0.52541719049994347348989373108, 2.58220530720247939426558635724, 3.16178586944181301205431547863, 4.61451895698113770553073896832, 5.38399753818842131486690992235, 6.69685529098953609003238537245, 7.34820877987022485272968946563, 7.70096107092951360246771479819, 9.173524617579037315858096752196, 10.19690908817264974558503912681, 10.73629048721790405204082572241

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