Properties

Label 2-690-23.22-c2-0-14
Degree $2$
Conductor $690$
Sign $0.774 + 0.632i$
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 − 1.52i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s + 2.20i·11-s − 3.46·12-s − 1.82·13-s + 2.15i·14-s − 3.87i·15-s + 4.00·16-s − 23.2i·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 − 0.217i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s + 0.200i·11-s − 0.288·12-s − 0.140·13-s + 0.153i·14-s − 0.258i·15-s + 0.250·16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.774 + 0.632i$
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: \((2,\ 690,\ (\ :1),\ 0.774 + 0.632i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8669152318\)
\(L(\frac12)\) \(\approx\) \(0.8669152318\)
\(L(2)\) 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 \( 1 + (14.5 - 17.8i)T \)
good7 \( 1 + 1.52iT - 49T^{2} \)
11 \( 1 - 2.20iT - 121T^{2} \)
13 \( 1 + 1.82T + 169T^{2} \)
17 \( 1 + 23.2iT - 289T^{2} \)
19 \( 1 - 2.85iT - 361T^{2} \)
29 \( 1 + 7.23T + 841T^{2} \)
31 \( 1 - 49.9T + 961T^{2} \)
37 \( 1 - 9.68iT - 1.36e3T^{2} \)
41 \( 1 + 62.1T + 1.68e3T^{2} \)
43 \( 1 + 76.6iT - 1.84e3T^{2} \)
47 \( 1 - 17.1T + 2.20e3T^{2} \)
53 \( 1 - 33.9iT - 2.80e3T^{2} \)
59 \( 1 - 35.2T + 3.48e3T^{2} \)
61 \( 1 - 8.54iT - 3.72e3T^{2} \)
67 \( 1 + 78.9iT - 4.48e3T^{2} \)
71 \( 1 - 111.T + 5.04e3T^{2} \)
73 \( 1 - 134.T + 5.32e3T^{2} \)
79 \( 1 + 72.4iT - 6.24e3T^{2} \)
83 \( 1 + 119. iT - 6.88e3T^{2} \)
89 \( 1 - 73.9iT - 7.92e3T^{2} \)
97 \( 1 + 178. iT - 9.40e3T^{2} \)
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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

−10.11205369382562532038059919158, −9.554550785845657678816809908432, −8.431956085509169822402465438992, −7.41310939772721801559036471388, −6.85761091826139122707899847477, −5.82798367261639216618467314863, −4.78641925018093022882457596693, −3.42429445709044359034393083092, −2.10267605197188014379791681504, −0.55253120880504243745400735495, 0.935851817101448739794560123645, 2.27294806911019681147223713478, 3.85629247157500504827555506444, 5.01671396563295211626890595889, 6.07075456842639870887492539000, 6.72946003618291675902930007414, 8.055335596681974442842807755886, 8.484324762623791980282735274053, 9.619107492901950472766097756899, 10.28660484929789224163199391556

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