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Properties

Label 2-702-39.5-c1-0-9
Degree $2$
Conductor $702$
Sign $0.747 - 0.663i$
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 + (3.09 + 3.09i)5-s + (2.51 + 2.51i)7-s + (0.707 − 0.707i)8-s − 4.37i·10-s + (1.30 − 1.30i)11-s + (3.38 + 1.24i)13-s − 3.55i·14-s − 1.00·16-s − 7.96·17-s + (1.81 − 1.81i)19-s + (−3.09 + 3.09i)20-s − 1.83·22-s − 4.57·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (1.38 + 1.38i)5-s + (0.949 + 0.949i)7-s + (0.250 − 0.250i)8-s − 1.38i·10-s + (0.392 − 0.392i)11-s + (0.938 + 0.346i)13-s − 0.949i·14-s − 0.250·16-s − 1.93·17-s + (0.416 − 0.416i)19-s + (−0.691 + 0.691i)20-s − 0.392·22-s − 0.954·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.747 - 0.663i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.747 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52796 + 0.580420i\)
\(L(\frac12)\) \(\approx\) \(1.52796 + 0.580420i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.38 - 1.24i)T \)
good5 \( 1 + (-3.09 - 3.09i)T + 5iT^{2} \)
7 \( 1 + (-2.51 - 2.51i)T + 7iT^{2} \)
11 \( 1 + (-1.30 + 1.30i)T - 11iT^{2} \)
17 \( 1 + 7.96T + 17T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 + 4.57T + 23T^{2} \)
29 \( 1 + 2.20iT - 29T^{2} \)
31 \( 1 + (-1.85 + 1.85i)T - 31iT^{2} \)
37 \( 1 + (2.39 + 2.39i)T + 37iT^{2} \)
41 \( 1 + (2.35 + 2.35i)T + 41iT^{2} \)
43 \( 1 + 6.20iT - 43T^{2} \)
47 \( 1 + (2.43 - 2.43i)T - 47iT^{2} \)
53 \( 1 + 8.05iT - 53T^{2} \)
59 \( 1 + (-9.17 + 9.17i)T - 59iT^{2} \)
61 \( 1 + 6.76T + 61T^{2} \)
67 \( 1 + (-4.76 + 4.76i)T - 67iT^{2} \)
71 \( 1 + (-7.32 - 7.32i)T + 71iT^{2} \)
73 \( 1 + (-5.48 - 5.48i)T + 73iT^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 + (-7.25 - 7.25i)T + 83iT^{2} \)
89 \( 1 + (-4.33 + 4.33i)T - 89iT^{2} \)
97 \( 1 + (-3.13 + 3.13i)T - 97iT^{2} \)
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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

−10.68353868500336382650946660262, −9.663205071835046060760610276781, −8.988518113811479419701835226173, −8.237799426473348098411907311011, −6.83911035271743039206390629488, −6.28351573129571724916905354150, −5.26753557106014613207704300344, −3.73971689565266401457989537195, −2.37373764345493484397546611577, −1.90721041760434689305410043506, 1.12850555057758015229686655069, 1.91554529049869622832799579576, 4.27473488603013439675244570995, 4.91685868908074216122233416966, 5.95238252741821278130290815099, 6.72992957480503814619574378171, 8.009638626561743536273508978484, 8.629565681832592424737022488291, 9.336422161421721051229313510980, 10.19693918148644169526254589700

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