Properties

Label 2-690-5.4-c1-0-11
Degree $2$
Conductor $690$
Sign $0.948 - 0.316i$
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 i·3-s − 4-s + (0.707 + 2.12i)5-s + 6-s − 2i·7-s i·8-s − 9-s + (−2.12 + 0.707i)10-s + 4.24·11-s + i·12-s − 4.82i·13-s + 2·14-s + (2.12 − 0.707i)15-s + 16-s + 1.17i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.316 + 0.948i)5-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.670 + 0.223i)10-s + 1.27·11-s + 0.288i·12-s − 1.33i·13-s + 0.534·14-s + (0.547 − 0.182i)15-s + 0.250·16-s + 0.284i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58258 + 0.256818i\)
\(L(\frac12)\) \(\approx\) \(1.58258 + 0.256818i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-0.707 - 2.12i)T \)
23 \( 1 - iT \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + 4.82iT - 13T^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3.41iT - 37T^{2} \)
41 \( 1 + 1.17T + 41T^{2} \)
43 \( 1 + 1.75iT - 43T^{2} \)
47 \( 1 - 4.82iT - 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 3.41T + 61T^{2} \)
67 \( 1 + 0.585iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 3.65iT - 73T^{2} \)
79 \( 1 + 7.65T + 79T^{2} \)
83 \( 1 - 1.41iT - 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−10.32087799143558851942097865662, −9.772163455233306341646731861350, −8.500884590332753668110169663426, −7.75245094452659092138341020588, −6.83621072564704550469791702038, −6.39984659361714939097998460042, −5.35001672789286272103546967679, −3.95168773788431369212375910229, −2.91712678565351579783689073692, −1.09905093290572192911390018785, 1.28810364690742197723188578930, 2.58579478179252669497453466095, 4.04468690662959987181201492212, 4.67086208995320879786151109404, 5.71823973354859464208238041088, 6.71683285618193204711957169424, 8.333299542889023975625281667380, 9.038493301593856911543272534412, 9.415932115066270440448047042429, 10.27693021387199110905159303060

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