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Properties

Label 2-690-69.68-c1-0-20
Degree $2$
Conductor $690$
Sign $0.941 + 0.336i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + i·2-s + (−0.798 − 1.53i)3-s − 4-s + 5-s + (1.53 − 0.798i)6-s − 0.145i·7-s − i·8-s + (−1.72 + 2.45i)9-s + i·10-s + 3.35·11-s + (0.798 + 1.53i)12-s + 2.96·13-s + 0.145·14-s + (−0.798 − 1.53i)15-s + 16-s − 5.54·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.461 − 0.887i)3-s − 0.5·4-s + 0.447·5-s + (0.627 − 0.326i)6-s − 0.0548i·7-s − 0.353i·8-s + (−0.574 + 0.818i)9-s + 0.316i·10-s + 1.01·11-s + (0.230 + 0.443i)12-s + 0.821·13-s + 0.0388·14-s + (−0.206 − 0.396i)15-s + 0.250·16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32323 - 0.229487i\)
\(L(\frac12)\) \(\approx\) \(1.32323 - 0.229487i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.798 + 1.53i)T \)
5 \( 1 - T \)
23 \( 1 + (-3.26 - 3.51i)T \)
good7 \( 1 + 0.145iT - 7T^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 + 6.91iT - 19T^{2} \)
29 \( 1 + 3.58iT - 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 + 6.04iT - 37T^{2} \)
41 \( 1 + 8.84iT - 41T^{2} \)
43 \( 1 + 8.40iT - 43T^{2} \)
47 \( 1 - 9.98iT - 47T^{2} \)
53 \( 1 - 3.83T + 53T^{2} \)
59 \( 1 - 7.91iT - 59T^{2} \)
61 \( 1 + 4.14iT - 61T^{2} \)
67 \( 1 + 13.0iT - 67T^{2} \)
71 \( 1 - 1.26iT - 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 + 9.71T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−10.56232452785273664468345895751, −9.086166887940832637056660927134, −8.828930908910241354005436533836, −7.54586449944151578092488127100, −6.72015930480692655230169405336, −6.25814638723712192762472738730, −5.22600384415739770041158013308, −4.15101678554830287376374772115, −2.43802200237516426368025860865, −0.912370895745096603589070550176, 1.30487855337306275813699494218, 2.94551198921606665679306465459, 4.04382726934257210141721822431, 4.76826663933484123331660209804, 6.03041076618595379592177657128, 6.59628897117612545922812155210, 8.467696470889214607926576913877, 8.895503491912234296366901410125, 9.958442805794588258473539717194, 10.38024697759217890395910001088

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