Properties

Label 2-690-115.53-c1-0-19
Degree $2$
Conductor $690$
Sign $-0.822 + 0.569i$
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  + (−0.0713 − 0.997i)2-s + (−0.977 − 0.212i)3-s + (−0.989 + 0.142i)4-s + (−0.0118 − 2.23i)5-s + (−0.142 + 0.989i)6-s + (1.21 + 0.663i)7-s + (0.212 + 0.977i)8-s + (0.909 + 0.415i)9-s + (−2.22 + 0.171i)10-s + (3.43 − 2.97i)11-s + (0.997 + 0.0713i)12-s + (−0.0572 − 0.104i)13-s + (0.574 − 1.25i)14-s + (−0.463 + 2.18i)15-s + (0.959 − 0.281i)16-s + (2.58 − 1.93i)17-s + ⋯
L(s)  = 1  + (−0.0504 − 0.705i)2-s + (−0.564 − 0.122i)3-s + (−0.494 + 0.0711i)4-s + (−0.00529 − 0.999i)5-s + (−0.0580 + 0.404i)6-s + (0.459 + 0.250i)7-s + (0.0751 + 0.345i)8-s + (0.303 + 0.138i)9-s + (−0.705 + 0.0541i)10-s + (1.03 − 0.897i)11-s + (0.287 + 0.0205i)12-s + (−0.0158 − 0.0290i)13-s + (0.153 − 0.336i)14-s + (−0.119 + 0.564i)15-s + (0.239 − 0.0704i)16-s + (0.626 − 0.469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.822 + 0.569i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331136 - 1.06011i\)
\(L(\frac12)\) \(\approx\) \(0.331136 - 1.06011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0713 + 0.997i)T \)
3 \( 1 + (0.977 + 0.212i)T \)
5 \( 1 + (0.0118 + 2.23i)T \)
23 \( 1 + (4.43 + 1.83i)T \)
good7 \( 1 + (-1.21 - 0.663i)T + (3.78 + 5.88i)T^{2} \)
11 \( 1 + (-3.43 + 2.97i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.0572 + 0.104i)T + (-7.02 + 10.9i)T^{2} \)
17 \( 1 + (-2.58 + 1.93i)T + (4.78 - 16.3i)T^{2} \)
19 \( 1 + (0.165 + 1.15i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-2.45 - 0.352i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (5.49 + 3.52i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-0.721 + 0.269i)T + (27.9 - 24.2i)T^{2} \)
41 \( 1 + (1.09 + 2.40i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (1.03 - 4.74i)T + (-39.1 - 17.8i)T^{2} \)
47 \( 1 + (3.35 - 3.35i)T - 47iT^{2} \)
53 \( 1 + (-2.37 + 4.35i)T + (-28.6 - 44.5i)T^{2} \)
59 \( 1 + (0.267 - 0.912i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-6.89 + 10.7i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (14.4 - 1.03i)T + (66.3 - 9.53i)T^{2} \)
71 \( 1 + (-5.17 + 5.96i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (6.67 - 8.91i)T + (-20.5 - 70.0i)T^{2} \)
79 \( 1 + (-11.2 - 3.31i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-0.511 - 1.37i)T + (-62.7 + 54.3i)T^{2} \)
89 \( 1 + (-7.15 + 4.59i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-4.90 + 13.1i)T + (-73.3 - 63.5i)T^{2} \)
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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

−10.13397434382190376723454622485, −9.304916863777630517743702528214, −8.564217597773326284447083394733, −7.74990388385831246796389460261, −6.32842457034891563375144325817, −5.44769423603255194771380935343, −4.58162583543244287163995785941, −3.55474445497577344527778722143, −1.86626903137632956100836654935, −0.68899763399899170138065285488, 1.68179489591330909608170576485, 3.58236190278400727649871153129, 4.44477748917135539880929158159, 5.63579890949928847282784992882, 6.45755776422316795733174688827, 7.20458395130177828955822591241, 7.923786797080398158159254852757, 9.151438303264954256629131607258, 10.07005240926852826787409184672, 10.59950265739852612826425327854

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