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Properties

Label 2-690-5.2-c2-0-38
Degree $2$
Conductor $690$
Sign $0.269 + 0.963i$
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 + (−2.41 − 4.37i)5-s + 2.44·6-s + (0.381 + 0.381i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (1.96 − 6.79i)10-s + 6.70·11-s + (2.44 + 2.44i)12-s + (3.27 − 3.27i)13-s + 0.763i·14-s + (−8.32 − 2.40i)15-s − 4·16-s + (−12.9 − 12.9i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.483 − 0.875i)5-s + 0.408·6-s + (0.0545 + 0.0545i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.196 − 0.679i)10-s + 0.609·11-s + (0.204 + 0.204i)12-s + (0.252 − 0.252i)13-s + 0.0545i·14-s + (−0.554 − 0.160i)15-s − 0.250·16-s + (−0.763 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.269 + 0.963i$
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: \((2,\ 690,\ (\ :1),\ 0.269 + 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.129854990\)
\(L(\frac12)\) \(\approx\) \(2.129854990\)
\(L(2)\) 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 \( 1 + (2.41 + 4.37i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (-0.381 - 0.381i)T + 49iT^{2} \)
11 \( 1 - 6.70T + 121T^{2} \)
13 \( 1 + (-3.27 + 3.27i)T - 169iT^{2} \)
17 \( 1 + (12.9 + 12.9i)T + 289iT^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
29 \( 1 + 43.2iT - 841T^{2} \)
31 \( 1 - 14.8T + 961T^{2} \)
37 \( 1 + (7.77 + 7.77i)T + 1.36e3iT^{2} \)
41 \( 1 + 19.9T + 1.68e3T^{2} \)
43 \( 1 + (-41.2 + 41.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (31.7 + 31.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-13.9 + 13.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 2.31iT - 3.48e3T^{2} \)
61 \( 1 - 22.8T + 3.72e3T^{2} \)
67 \( 1 + (-59.1 - 59.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 29.7T + 5.04e3T^{2} \)
73 \( 1 + (73.2 - 73.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 12.0iT - 6.24e3T^{2} \)
83 \( 1 + (12.8 - 12.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 42.0iT - 7.92e3T^{2} \)
97 \( 1 + (-8.47 - 8.47i)T + 9.40e3iT^{2} \)
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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

−9.827597541085834019093590750267, −8.880769012506943147136815202426, −8.390841784629057977130008063536, −7.39121926813497234630126413025, −6.65843532500349196162252568122, −5.51851537002875786337392299638, −4.54430350403605956883456410486, −3.68553232715157024260457947864, −2.30060273423773960585078719590, −0.61206761322573380914735252725, 1.67496897677491803685133689782, 2.98987568575545062288715091141, 3.82149979274264959411226488120, 4.57968506516516199250919214222, 6.01532255383786253102773528041, 6.76970167687003086499181007322, 7.902972077505592143132458198569, 8.810442026469355308137136450131, 9.775098727143484340476287726867, 10.64894255666613333403394393680

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