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Properties

Label 2-702-39.5-c1-0-18
Degree $2$
Conductor $702$
Sign $-0.502 - 0.864i$
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 + (−2.82 − 2.82i)5-s + (−1.76 − 1.76i)7-s + (0.707 − 0.707i)8-s + 4.00i·10-s + (4.00 − 4.00i)11-s + (−1.89 + 3.06i)13-s + 2.50i·14-s − 1.00·16-s − 3.07·17-s + (1.61 − 1.61i)19-s + (2.82 − 2.82i)20-s − 5.66·22-s − 2.77·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.26 − 1.26i)5-s + (−0.668 − 0.668i)7-s + (0.250 − 0.250i)8-s + 1.26i·10-s + (1.20 − 1.20i)11-s + (−0.525 + 0.850i)13-s + 0.668i·14-s − 0.250·16-s − 0.745·17-s + (0.370 − 0.370i)19-s + (0.632 − 0.632i)20-s − 1.20·22-s − 0.577·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.502 - 0.864i$
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: \((2,\ 702,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0978031 + 0.169919i\)
\(L(\frac12)\) \(\approx\) \(0.0978031 + 0.169919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 - 3.06i)T \)
good5 \( 1 + (2.82 + 2.82i)T + 5iT^{2} \)
7 \( 1 + (1.76 + 1.76i)T + 7iT^{2} \)
11 \( 1 + (-4.00 + 4.00i)T - 11iT^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 + (-1.61 + 1.61i)T - 19iT^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 - 9.35iT - 29T^{2} \)
31 \( 1 + (2.23 - 2.23i)T - 31iT^{2} \)
37 \( 1 + (-1.94 - 1.94i)T + 37iT^{2} \)
41 \( 1 + (7.64 + 7.64i)T + 41iT^{2} \)
43 \( 1 + 1.66iT - 43T^{2} \)
47 \( 1 + (6.59 - 6.59i)T - 47iT^{2} \)
53 \( 1 + 7.68iT - 53T^{2} \)
59 \( 1 + (1.49 - 1.49i)T - 59iT^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 + (8.71 - 8.71i)T - 67iT^{2} \)
71 \( 1 + (-6.55 - 6.55i)T + 71iT^{2} \)
73 \( 1 + (2.06 + 2.06i)T + 73iT^{2} \)
79 \( 1 - 9.13T + 79T^{2} \)
83 \( 1 + (1.85 + 1.85i)T + 83iT^{2} \)
89 \( 1 + (-2.39 + 2.39i)T - 89iT^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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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

−9.690577596451242638996642761426, −8.850150190213647504218177707954, −8.554935924922601860924566030066, −7.32059879411748289618568378971, −6.63383832807257239687484566367, −5.01094687290851250288514020672, −4.00124877996291415384977729725, −3.42448839956512083417775433650, −1.35238195687995065442121327365, −0.12925260469411471110938494911, 2.32990198605598681149527381062, 3.53650556789905484899314303776, 4.55198800771072693455772127990, 6.11875318073892218035247709985, 6.71201658688016944650794513085, 7.54934666256036501874876106147, 8.171826013437739651219618168773, 9.467177612129241889532128682391, 9.933574368788880183851722863684, 10.95346666665418299704759581986

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