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Properties

Label 2-690-69.68-c1-0-27
Degree $2$
Conductor $690$
Sign $-0.0157 + 0.999i$
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 + (1.05 − 1.37i)3-s − 4-s − 5-s + (1.37 + 1.05i)6-s − 1.91i·7-s − i·8-s + (−0.772 − 2.89i)9-s − i·10-s − 2.07·11-s + (−1.05 + 1.37i)12-s + 1.28·13-s + 1.91·14-s + (−1.05 + 1.37i)15-s + 16-s − 4.04·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.609 − 0.792i)3-s − 0.5·4-s − 0.447·5-s + (0.560 + 0.430i)6-s − 0.722i·7-s − 0.353i·8-s + (−0.257 − 0.966i)9-s − 0.316i·10-s − 0.625·11-s + (−0.304 + 0.396i)12-s + 0.357·13-s + 0.510·14-s + (−0.272 + 0.354i)15-s + 0.250·16-s − 0.981·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0157 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0157 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0157 + 0.999i$
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: \((2,\ 690,\ (\ :1/2),\ -0.0157 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804103 - 0.816878i\)
\(L(\frac12)\) \(\approx\) \(0.804103 - 0.816878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 + 1.37i)T \)
5 \( 1 + T \)
23 \( 1 + (2.86 + 3.84i)T \)
good7 \( 1 + 1.91iT - 7T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 + 3.79iT - 19T^{2} \)
29 \( 1 + 2.78iT - 29T^{2} \)
31 \( 1 - 1.87T + 31T^{2} \)
37 \( 1 + 2.18iT - 37T^{2} \)
41 \( 1 - 0.590iT - 41T^{2} \)
43 \( 1 - 0.332iT - 43T^{2} \)
47 \( 1 + 4.11iT - 47T^{2} \)
53 \( 1 - 3.47T + 53T^{2} \)
59 \( 1 + 2.07iT - 59T^{2} \)
61 \( 1 - 3.23iT - 61T^{2} \)
67 \( 1 + 6.73iT - 67T^{2} \)
71 \( 1 - 9.09iT - 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 + 3.38iT - 79T^{2} \)
83 \( 1 - 5.68T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−10.15299159806571530173947772447, −9.008684792985016777231396968351, −8.366712025724776312061447488448, −7.57088658258026442525780944682, −6.90166378893785594352778807922, −6.09657087455189152520999544867, −4.68857994146194287757305687279, −3.74478747131969800238908511911, −2.40451872985110826843879329468, −0.54815533732399856343745047173, 2.02279856203242805409401885159, 3.08462452567991313939193683588, 4.00904559631332258850930364938, 4.96837999171065860796215785902, 5.95039872828841927499015814278, 7.55308574010651795813090078099, 8.404459592925919019340143256788, 8.989043869329282853416550251264, 9.902496487607335776874074482136, 10.63494517965109365711267420720

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