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Properties

Label 2-702-117.11-c1-0-4
Degree $2$
Conductor $702$
Sign $0.838 + 0.544i$
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 + (−0.650 − 0.174i)5-s + (−1.73 − 0.463i)7-s + (0.707 + 0.707i)8-s + (0.583 − 0.336i)10-s + (−0.696 − 0.696i)11-s + (0.391 + 3.58i)13-s + (1.55 − 0.895i)14-s − 1.00·16-s + (1.20 − 2.09i)17-s + (6.33 − 1.69i)19-s + (−0.174 + 0.650i)20-s + 0.985·22-s + (3.16 − 5.47i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.290 − 0.0779i)5-s + (−0.654 − 0.175i)7-s + (0.250 + 0.250i)8-s + (0.184 − 0.106i)10-s + (−0.210 − 0.210i)11-s + (0.108 + 0.994i)13-s + (0.414 − 0.239i)14-s − 0.250·16-s + (0.293 − 0.508i)17-s + (1.45 − 0.389i)19-s + (−0.0389 + 0.145i)20-s + 0.210·22-s + (0.659 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.838 + 0.544i$
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: \((2,\ 702,\ (\ :1/2),\ 0.838 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868198 - 0.257336i\)
\(L(\frac12)\) \(\approx\) \(0.868198 - 0.257336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.391 - 3.58i)T \)
good5 \( 1 + (0.650 + 0.174i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.73 + 0.463i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.696 + 0.696i)T + 11iT^{2} \)
17 \( 1 + (-1.20 + 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.33 + 1.69i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.16 + 5.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.29iT - 29T^{2} \)
31 \( 1 + (-1.49 + 5.59i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.83 - 1.83i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.28 + 4.80i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.772 - 0.446i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.66 + 0.981i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 9.00iT - 53T^{2} \)
59 \( 1 + (-3.33 - 3.33i)T + 59iT^{2} \)
61 \( 1 + (2.38 + 4.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 2.86i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.70 - 6.37i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.01 - 8.01i)T - 73iT^{2} \)
79 \( 1 + (0.807 - 1.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.95 - 7.31i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.440 + 1.64i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.497 - 1.85i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.995700217170875141733922237889, −9.583510225703115236611811553739, −8.625491512060322126848795320241, −7.72612692714874308246914562119, −6.92103768156212317158514035147, −6.11909245683144153072429496272, −5.00816965958038622547819531112, −3.89711915312585493655383301484, −2.52646792685621538639117578287, −0.64489639809985541176037089981, 1.25990990915106581760490901307, 2.97922713343122102028904199605, 3.57348114456136974051642166162, 5.10895859515114944174575611424, 6.05954095544526853171308427427, 7.38462246764129877029892825981, 7.84458396521967690953602823921, 8.995953318232480185611914971065, 9.728984051644047042795210171328, 10.41553987805536627679315255202

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