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Properties

Label 2-690-5.2-c2-0-22
Degree $2$
Conductor $690$
Sign $0.971 - 0.238i$
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 + (−4.45 + 2.27i)5-s + 2.44·6-s + (−3.57 − 3.57i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−6.72 − 2.17i)10-s + 15.2·11-s + (2.44 + 2.44i)12-s + (9.30 − 9.30i)13-s − 7.15i·14-s + (−2.66 + 8.23i)15-s − 4·16-s + (7.64 + 7.64i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.890 + 0.455i)5-s + 0.408·6-s + (−0.510 − 0.510i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.672 − 0.217i)10-s + 1.38·11-s + (0.204 + 0.204i)12-s + (0.715 − 0.715i)13-s − 0.510i·14-s + (−0.177 + 0.549i)15-s − 0.250·16-s + (0.449 + 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.971 - 0.238i$
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: \((2,\ 690,\ (\ :1),\ 0.971 - 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.465881639\)
\(L(\frac12)\) \(\approx\) \(2.465881639\)
\(L(2)\) 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 \( 1 + (4.45 - 2.27i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (3.57 + 3.57i)T + 49iT^{2} \)
11 \( 1 - 15.2T + 121T^{2} \)
13 \( 1 + (-9.30 + 9.30i)T - 169iT^{2} \)
17 \( 1 + (-7.64 - 7.64i)T + 289iT^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
29 \( 1 - 28.4iT - 841T^{2} \)
31 \( 1 - 46.8T + 961T^{2} \)
37 \( 1 + (-38.8 - 38.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 49.1T + 1.68e3T^{2} \)
43 \( 1 + (39.9 - 39.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-42.2 - 42.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-40.9 + 40.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 115. iT - 3.48e3T^{2} \)
61 \( 1 + 39.5T + 3.72e3T^{2} \)
67 \( 1 + (-30.0 - 30.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 109.T + 5.04e3T^{2} \)
73 \( 1 + (-37.5 + 37.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 116. iT - 6.24e3T^{2} \)
83 \( 1 + (-97.5 + 97.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 26.3iT - 7.92e3T^{2} \)
97 \( 1 + (70.0 + 70.0i)T + 9.40e3iT^{2} \)
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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

−10.35307399897890760942461943341, −9.230164392402365417643151751447, −8.299606690418847860589589439687, −7.62066872744501040822861480141, −6.64708878714824221034156726061, −6.24828309379331131848542609479, −4.59559041660013928716608893963, −3.66382341686241941768107047352, −3.00321742402291963921514756559, −0.958274854840474217040091513538, 1.09635314657912325985895665963, 2.66189533034478168604586545792, 4.00311023529509961584205648338, 4.11357819354495990926143682608, 5.62833975491921988251410112282, 6.51886708747418136986103945194, 7.73427420502945175319039074275, 8.816644939915540734906972277354, 9.276539901692413039580172520416, 10.21349504236377797466736194191

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