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Properties

Label 2-690-5.2-c2-0-42
Degree $2$
Conductor $690$
Sign $-0.179 - 0.983i$
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 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (−2.00 − 4.58i)5-s − 2.44·6-s + (−5.25 − 5.25i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−2.57 + 6.58i)10-s + 3.64·11-s + (2.44 + 2.44i)12-s + (−7.35 + 7.35i)13-s + 10.5i·14-s + (−8.06 − 3.15i)15-s − 4·16-s + (−14.6 − 14.6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.400 − 0.916i)5-s − 0.408·6-s + (−0.750 − 0.750i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.257 + 0.658i)10-s + 0.331·11-s + (0.204 + 0.204i)12-s + (−0.565 + 0.565i)13-s + 0.750i·14-s + (−0.537 − 0.210i)15-s − 0.250·16-s + (−0.862 − 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.179 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2079624723\)
\(L(\frac12)\) \(\approx\) \(0.2079624723\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (2.00 + 4.58i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (5.25 + 5.25i)T + 49iT^{2} \)
11 \( 1 - 3.64T + 121T^{2} \)
13 \( 1 + (7.35 - 7.35i)T - 169iT^{2} \)
17 \( 1 + (14.6 + 14.6i)T + 289iT^{2} \)
19 \( 1 - 3.37iT - 361T^{2} \)
29 \( 1 + 1.23iT - 841T^{2} \)
31 \( 1 - 19.8T + 961T^{2} \)
37 \( 1 + (6.16 + 6.16i)T + 1.36e3iT^{2} \)
41 \( 1 - 47.2T + 1.68e3T^{2} \)
43 \( 1 + (35.5 - 35.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.2 - 33.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (26.6 - 26.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 44.0iT - 3.48e3T^{2} \)
61 \( 1 + 37.9T + 3.72e3T^{2} \)
67 \( 1 + (87.7 + 87.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 7.77T + 5.04e3T^{2} \)
73 \( 1 + (58.7 - 58.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 26.9iT - 6.24e3T^{2} \)
83 \( 1 + (1.79 - 1.79i)T - 6.88e3iT^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 + (-1.22 - 1.22i)T + 9.40e3iT^{2} \)
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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

−9.352390489704705807088581994357, −9.091611138837198727070024582850, −7.919814987506441698835047786784, −7.26205095411621286399975184455, −6.33343900179517122317980005078, −4.69886795097641708160297665497, −3.90795373711456479329049101992, −2.68974578555797342358983636146, −1.28805274301259441851993219516, −0.085968525027249274852276190627, 2.27102470486301699809395333614, 3.25014935136020878109438122388, 4.43080896945775160152305921316, 5.79363035938132546865969573572, 6.55214315410726003429395552340, 7.42650102817675005958923722647, 8.384233152453717836348718129532, 9.085138270541337367259143961420, 10.00844834962875202524087438914, 10.55904365870125696815608764770

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