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Properties

Label 2-690-115.114-c2-0-40
Degree $2$
Conductor $690$
Sign $0.834 + 0.551i$
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 + (4.95 + 0.671i)5-s + 2.44·6-s + 12.0·7-s − 2.82i·8-s − 2.99·9-s + (−0.949 + 7.00i)10-s − 20.5i·11-s + 3.46i·12-s − 13.6i·13-s + 17.1i·14-s + (1.16 − 8.58i)15-s + 4.00·16-s + 0.319·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.990 + 0.134i)5-s + 0.408·6-s + 1.72·7-s − 0.353i·8-s − 0.333·9-s + (−0.0949 + 0.700i)10-s − 1.86i·11-s + 0.288i·12-s − 1.05i·13-s + 1.22i·14-s + (0.0775 − 0.572i)15-s + 0.250·16-s + 0.0187·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.834 + 0.551i$
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: \((2,\ 690,\ (\ :1),\ 0.834 + 0.551i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.347315789\)
\(L(\frac12)\) \(\approx\) \(2.347315789\)
\(L(2)\) 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 \( 1 + (-4.95 - 0.671i)T \)
23 \( 1 + (20.7 + 9.98i)T \)
good7 \( 1 - 12.0T + 49T^{2} \)
11 \( 1 + 20.5iT - 121T^{2} \)
13 \( 1 + 13.6iT - 169T^{2} \)
17 \( 1 - 0.319T + 289T^{2} \)
19 \( 1 - 6.05iT - 361T^{2} \)
29 \( 1 + 6.76T + 841T^{2} \)
31 \( 1 + 30.6T + 961T^{2} \)
37 \( 1 + 71.2T + 1.36e3T^{2} \)
41 \( 1 - 12.7T + 1.68e3T^{2} \)
43 \( 1 - 0.0776T + 1.84e3T^{2} \)
47 \( 1 + 35.9iT - 2.20e3T^{2} \)
53 \( 1 - 11.2T + 2.80e3T^{2} \)
59 \( 1 - 47.3T + 3.48e3T^{2} \)
61 \( 1 - 42.0iT - 3.72e3T^{2} \)
67 \( 1 - 75.9T + 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 - 107. iT - 5.32e3T^{2} \)
79 \( 1 + 68.2iT - 6.24e3T^{2} \)
83 \( 1 - 103.T + 6.88e3T^{2} \)
89 \( 1 - 50.8iT - 7.92e3T^{2} \)
97 \( 1 - 39.8T + 9.40e3T^{2} \)
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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

−10.32973426546011075242923908915, −8.885375466101626278593150673323, −8.358346445156467757212232849543, −7.70830782717852915693583634033, −6.56229142480594279827110578026, −5.50761577349370760238939907254, −5.34781286778690738225838805615, −3.60724658303711197799581711705, −2.11910038765668654669693082623, −0.871519662319826348039213974005, 1.75899198309389001499814325514, 2.06990471730639263427271267964, 4.00124930805811594863395741321, 4.82076556568209844952374666645, 5.34988020838941163043264768558, 6.86463289804419373112692568894, 7.914850313984331875323204990275, 8.998264440729333599482453484910, 9.545626219496867555813234316916, 10.35901297691369310310414550521

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