Properties

Label 2-690-23.22-c2-0-27
Degree $2$
Conductor $690$
Sign $-0.909 + 0.414i$
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  − 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s − 2.44·6-s − 3.49i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s + 11.9i·11-s + 3.46·12-s − 19.9·13-s + 4.94i·14-s − 3.87i·15-s + 4.00·16-s − 26.2i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.447i·5-s − 0.408·6-s − 0.499i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s + 1.08i·11-s + 0.288·12-s − 1.53·13-s + 0.353i·14-s − 0.258i·15-s + 0.250·16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.909 + 0.414i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.909 + 0.414i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5564304952\)
\(L(\frac12)\) \(\approx\) \(0.5564304952\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (9.54 + 20.9i)T \)
good7 \( 1 + 3.49iT - 49T^{2} \)
11 \( 1 - 11.9iT - 121T^{2} \)
13 \( 1 + 19.9T + 169T^{2} \)
17 \( 1 + 26.2iT - 289T^{2} \)
19 \( 1 - 0.0297iT - 361T^{2} \)
29 \( 1 + 41.0T + 841T^{2} \)
31 \( 1 - 29.7T + 961T^{2} \)
37 \( 1 - 51.3iT - 1.36e3T^{2} \)
41 \( 1 + 74.6T + 1.68e3T^{2} \)
43 \( 1 + 51.7iT - 1.84e3T^{2} \)
47 \( 1 + 27.5T + 2.20e3T^{2} \)
53 \( 1 + 15.0iT - 2.80e3T^{2} \)
59 \( 1 - 1.55T + 3.48e3T^{2} \)
61 \( 1 + 12.9iT - 3.72e3T^{2} \)
67 \( 1 - 28.2iT - 4.48e3T^{2} \)
71 \( 1 + 94.7T + 5.04e3T^{2} \)
73 \( 1 + 25.3T + 5.32e3T^{2} \)
79 \( 1 + 80.5iT - 6.24e3T^{2} \)
83 \( 1 - 55.8iT - 6.88e3T^{2} \)
89 \( 1 + 81.9iT - 7.92e3T^{2} \)
97 \( 1 - 4.03iT - 9.40e3T^{2} \)
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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

−9.888755379628597137971215104336, −9.143876684418263197921452151075, −8.171799732912527906989303207821, −7.30400170580607736265905123490, −6.86590579013426075846505534680, −5.15739082912505662883806744010, −4.38990530059016399511438166486, −2.86811216376112318900534175998, −1.84526878595363270284225559831, −0.21424252648069691488196490117, 1.76836421142392285467031840928, 2.81764605480023598099551593092, 3.84534309759302869290938046481, 5.43759868943235839684502193145, 6.32496353016301323145149106407, 7.42156164642271720286433175435, 8.087136527542948769672894201619, 8.907851010730070085036157443732, 9.723397012858319932238806371794, 10.43452342614560844737661290978

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