Properties

Label 2-690-115.114-c2-0-7
Degree $2$
Conductor $690$
Sign $0.988 - 0.149i$
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 + (−4.51 − 2.15i)5-s − 2.44·6-s − 9.45·7-s + 2.82i·8-s − 2.99·9-s + (−3.04 + 6.38i)10-s + 0.168i·11-s + 3.46i·12-s − 0.410i·13-s + 13.3i·14-s + (−3.73 + 7.81i)15-s + 4.00·16-s − 1.40·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (−0.902 − 0.431i)5-s − 0.408·6-s − 1.35·7-s + 0.353i·8-s − 0.333·9-s + (−0.304 + 0.638i)10-s + 0.0153i·11-s + 0.288i·12-s − 0.0315i·13-s + 0.954i·14-s + (−0.248 + 0.520i)15-s + 0.250·16-s − 0.0827·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.988 - 0.149i$
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: \((2,\ 690,\ (\ :1),\ 0.988 - 0.149i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4882071028\)
\(L(\frac12)\) \(\approx\) \(0.4882071028\)
\(L(2)\) 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 \( 1 + (4.51 + 2.15i)T \)
23 \( 1 + (19.0 - 12.9i)T \)
good7 \( 1 + 9.45T + 49T^{2} \)
11 \( 1 - 0.168iT - 121T^{2} \)
13 \( 1 + 0.410iT - 169T^{2} \)
17 \( 1 + 1.40T + 289T^{2} \)
19 \( 1 + 13.0iT - 361T^{2} \)
29 \( 1 - 3.27T + 841T^{2} \)
31 \( 1 - 46.3T + 961T^{2} \)
37 \( 1 + 8.23T + 1.36e3T^{2} \)
41 \( 1 - 22.2T + 1.68e3T^{2} \)
43 \( 1 - 27.5T + 1.84e3T^{2} \)
47 \( 1 + 24.8iT - 2.20e3T^{2} \)
53 \( 1 - 5.90T + 2.80e3T^{2} \)
59 \( 1 + 69.0T + 3.48e3T^{2} \)
61 \( 1 - 106. iT - 3.72e3T^{2} \)
67 \( 1 + 19.7T + 4.48e3T^{2} \)
71 \( 1 - 10.3T + 5.04e3T^{2} \)
73 \( 1 - 8.92iT - 5.32e3T^{2} \)
79 \( 1 - 84.6iT - 6.24e3T^{2} \)
83 \( 1 - 60.8T + 6.88e3T^{2} \)
89 \( 1 + 9.16iT - 7.92e3T^{2} \)
97 \( 1 + 86.3T + 9.40e3T^{2} \)
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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

−10.30488650211857482052154247222, −9.438804866672657932370784142692, −8.666522342255847435417676481314, −7.74373396292429532852196884018, −6.82830611798659225796693778527, −5.82907936671698125074619663513, −4.53229869681112954812271428449, −3.54054211735500292557665419750, −2.59608839201577721091447271242, −0.907438374193276853782112228573, 0.23371587373484943245404692005, 2.89148665010381768742032066075, 3.78722421369822076573498893228, 4.62592079989023533195722718417, 6.04191395321247825304587044895, 6.56443391621902876315621591976, 7.64901971185435816587925822373, 8.404660547007251349207713023205, 9.417873081241967248139003922924, 10.11353531675452018329118673448

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