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Properties

Label 2-702-117.32-c1-0-7
Degree $2$
Conductor $702$
Sign $0.750 - 0.661i$
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 + (−0.339 + 0.0908i)5-s + (2.97 − 0.797i)7-s + (−0.707 + 0.707i)8-s + (−0.304 − 0.175i)10-s + (2.35 − 2.35i)11-s + (2.60 + 2.49i)13-s + (2.66 + 1.54i)14-s − 1.00·16-s + (−3.87 − 6.71i)17-s + (3.67 + 0.984i)19-s + (−0.0908 − 0.339i)20-s + 3.32·22-s + (3.34 + 5.78i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.151 + 0.0406i)5-s + (1.12 − 0.301i)7-s + (−0.250 + 0.250i)8-s + (−0.0961 − 0.0555i)10-s + (0.709 − 0.709i)11-s + (0.723 + 0.690i)13-s + (0.713 + 0.411i)14-s − 0.250·16-s + (−0.940 − 1.62i)17-s + (0.842 + 0.225i)19-s + (−0.0203 − 0.0758i)20-s + 0.709·22-s + (0.696 + 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.750 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09599 + 0.792034i\)
\(L(\frac12)\) \(\approx\) \(2.09599 + 0.792034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 2.49i)T \)
good5 \( 1 + (0.339 - 0.0908i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.97 + 0.797i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.35 + 2.35i)T - 11iT^{2} \)
17 \( 1 + (3.87 + 6.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.67 - 0.984i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.34 - 5.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.68iT - 29T^{2} \)
31 \( 1 + (-0.293 - 1.09i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-8.90 + 2.38i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.678 + 2.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.68 + 2.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.17 - 1.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (7.75 - 7.75i)T - 59iT^{2} \)
61 \( 1 + (-5.01 + 8.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.3 + 3.56i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.343 + 1.28i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.19 + 2.19i)T + 73iT^{2} \)
79 \( 1 + (6.22 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.05 + 3.92i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.43 + 5.37i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.218 + 0.816i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.94884838004601113201340970757, −9.313644688956026752286220197629, −8.880789722193051318450858162638, −7.66252207929772145850122250251, −7.16004758246433146748663721113, −6.02913561028382573797481976312, −5.07516997993696705601487679602, −4.20685791281675619221676282006, −3.15827227264436578004411660000, −1.42229370716947420612601577356, 1.34848634599719513861822936197, 2.49846309281504426862619964546, 4.00970281979773894905365489664, 4.59873381229569402728677702024, 5.78064442949038547878190102044, 6.59551975679610779493854504999, 7.952994516680609186358433088111, 8.539680443564244390660695187294, 9.626475466704547844596473613027, 10.55831090109355857556813434476

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