Properties

Label 2-690-115.114-c2-0-0
Degree $2$
Conductor $690$
Sign $-0.590 + 0.807i$
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 + (3.10 − 3.91i)5-s − 2.44·6-s − 3.09·7-s − 2.82i·8-s − 2.99·9-s + (5.53 + 4.39i)10-s − 15.9i·11-s − 3.46i·12-s + 24.8i·13-s − 4.37i·14-s + (6.78 + 5.38i)15-s + 4.00·16-s − 12.6·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.621 − 0.783i)5-s − 0.408·6-s − 0.442·7-s − 0.353i·8-s − 0.333·9-s + (0.553 + 0.439i)10-s − 1.44i·11-s − 0.288i·12-s + 1.91i·13-s − 0.312i·14-s + (0.452 + 0.359i)15-s + 0.250·16-s − 0.746·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.590 + 0.807i$
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: \((2,\ 690,\ (\ :1),\ -0.590 + 0.807i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05237754372\)
\(L(\frac12)\) \(\approx\) \(0.05237754372\)
\(L(2)\) 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 \( 1 + (-3.10 + 3.91i)T \)
23 \( 1 + (22.9 - 0.906i)T \)
good7 \( 1 + 3.09T + 49T^{2} \)
11 \( 1 + 15.9iT - 121T^{2} \)
13 \( 1 - 24.8iT - 169T^{2} \)
17 \( 1 + 12.6T + 289T^{2} \)
19 \( 1 - 10.4iT - 361T^{2} \)
29 \( 1 + 44.7T + 841T^{2} \)
31 \( 1 + 50.0T + 961T^{2} \)
37 \( 1 + 39.1T + 1.36e3T^{2} \)
41 \( 1 - 41.3T + 1.68e3T^{2} \)
43 \( 1 - 71.6T + 1.84e3T^{2} \)
47 \( 1 + 17.3iT - 2.20e3T^{2} \)
53 \( 1 + 39.6T + 2.80e3T^{2} \)
59 \( 1 - 24.0T + 3.48e3T^{2} \)
61 \( 1 - 76.0iT - 3.72e3T^{2} \)
67 \( 1 + 76.7T + 4.48e3T^{2} \)
71 \( 1 - 22.9T + 5.04e3T^{2} \)
73 \( 1 + 51.8iT - 5.32e3T^{2} \)
79 \( 1 - 25.5iT - 6.24e3T^{2} \)
83 \( 1 + 21.6T + 6.88e3T^{2} \)
89 \( 1 + 165. iT - 7.92e3T^{2} \)
97 \( 1 + 0.580T + 9.40e3T^{2} \)
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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

−10.74527145993959660905755061464, −9.536912142168441421722197707802, −9.120858261265047842235156865329, −8.517189969630824099489187012022, −7.26250657785708600801742860275, −6.07916369668718749983440359956, −5.70940301442540322747395559489, −4.43561798933618007544610426165, −3.68847217078965683664433874653, −1.88095029215512404465558505739, 0.01721758894031926635651381165, 1.84536395814976006000644083504, 2.65879446130180932374283007671, 3.75680847254546312803532228245, 5.22976933092622523479094004313, 6.06758943249352296495448220299, 7.18738965915942605461620382781, 7.78108200317360708464984166966, 9.182000729147710346899078612650, 9.777865263061424454454577621049

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