Properties

Label 2-690-115.114-c2-0-17
Degree $2$
Conductor $690$
Sign $0.706 + 0.708i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (−1.73 + 4.68i)5-s − 2.44·6-s − 7.48·7-s + 2.82i·8-s − 2.99·9-s + (6.63 + 2.45i)10-s − 2.31i·11-s + 3.46i·12-s − 6.23i·13-s + 10.5i·14-s + (8.12 + 3.00i)15-s + 4.00·16-s + 2.91·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (−0.346 + 0.937i)5-s − 0.408·6-s − 1.06·7-s + 0.353i·8-s − 0.333·9-s + (0.663 + 0.245i)10-s − 0.210i·11-s + 0.288i·12-s − 0.479i·13-s + 0.756i·14-s + (0.541 + 0.200i)15-s + 0.250·16-s + 0.171·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.706 + 0.708i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.706 + 0.708i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.247407847\)
\(L(\frac12)\) \(\approx\) \(1.247407847\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + 1.73iT \)
5 \( 1 + (1.73 - 4.68i)T \)
23 \( 1 + (-9.64 + 20.8i)T \)
good7 \( 1 + 7.48T + 49T^{2} \)
11 \( 1 + 2.31iT - 121T^{2} \)
13 \( 1 + 6.23iT - 169T^{2} \)
17 \( 1 - 2.91T + 289T^{2} \)
19 \( 1 - 25.2iT - 361T^{2} \)
29 \( 1 - 42.7T + 841T^{2} \)
31 \( 1 - 20.9T + 961T^{2} \)
37 \( 1 - 62.7T + 1.36e3T^{2} \)
41 \( 1 - 33.4T + 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 - 38.5iT - 2.20e3T^{2} \)
53 \( 1 + 31.9T + 2.80e3T^{2} \)
59 \( 1 - 22.5T + 3.48e3T^{2} \)
61 \( 1 + 17.0iT - 3.72e3T^{2} \)
67 \( 1 - 77.7T + 4.48e3T^{2} \)
71 \( 1 + 134.T + 5.04e3T^{2} \)
73 \( 1 + 4.53iT - 5.32e3T^{2} \)
79 \( 1 - 28.3iT - 6.24e3T^{2} \)
83 \( 1 + 19.1T + 6.88e3T^{2} \)
89 \( 1 - 18.2iT - 7.92e3T^{2} \)
97 \( 1 - 145.T + 9.40e3T^{2} \)
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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

−10.27098744496305790232891367042, −9.565751011225310927338425331275, −8.343981691850978827844577173564, −7.63380694684405152729139929225, −6.48541966609478986592082813168, −5.95085405258317769531248697214, −4.31160518060030715574112306040, −3.18023066172854347082772159838, −2.57828362228274458875776401370, −0.791103912910045336880979585557, 0.70557494061066383511989579997, 2.91888830312984773707740953705, 4.16260951316626587450233145676, 4.83416918894956081173341450847, 5.88817693687341670570456820017, 6.80638801621335055445894930968, 7.77991299968426431408911801273, 8.810650927247391868157398332514, 9.360308811259154699988321646081, 10.00384132239717701360937372458

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