Properties

Label 2-690-5.3-c2-0-39
Degree $2$
Conductor $690$
Sign $0.0181 + 0.999i$
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 + (1.06 − 4.88i)5-s + 2.44·6-s + (6.77 − 6.77i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−3.82 − 5.94i)10-s + 6.48·11-s + (2.44 − 2.44i)12-s + (7.70 + 7.70i)13-s − 13.5i·14-s + (7.28 − 4.68i)15-s − 4·16-s + (0.781 − 0.781i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.212 − 0.977i)5-s + 0.408·6-s + (0.968 − 0.968i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.382 − 0.594i)10-s + 0.589·11-s + (0.204 − 0.204i)12-s + (0.592 + 0.592i)13-s − 0.968i·14-s + (0.485 − 0.312i)15-s − 0.250·16-s + (0.0459 − 0.0459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0181 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0181 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0181 + 0.999i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.0181 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.253241544\)
\(L(\frac12)\) \(\approx\) \(3.253241544\)
\(L(2)\) 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 \( 1 + (-1.06 + 4.88i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-6.77 + 6.77i)T - 49iT^{2} \)
11 \( 1 - 6.48T + 121T^{2} \)
13 \( 1 + (-7.70 - 7.70i)T + 169iT^{2} \)
17 \( 1 + (-0.781 + 0.781i)T - 289iT^{2} \)
19 \( 1 - 0.529iT - 361T^{2} \)
29 \( 1 + 22.8iT - 841T^{2} \)
31 \( 1 + 24.3T + 961T^{2} \)
37 \( 1 + (-6.38 + 6.38i)T - 1.36e3iT^{2} \)
41 \( 1 - 16.5T + 1.68e3T^{2} \)
43 \( 1 + (-11.3 - 11.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.1 - 19.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (7.76 + 7.76i)T + 2.80e3iT^{2} \)
59 \( 1 + 62.7iT - 3.48e3T^{2} \)
61 \( 1 - 37.8T + 3.72e3T^{2} \)
67 \( 1 + (23.8 - 23.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 82.9T + 5.04e3T^{2} \)
73 \( 1 + (-32.9 - 32.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 55.5iT - 6.24e3T^{2} \)
83 \( 1 + (53.8 + 53.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 9.20iT - 7.92e3T^{2} \)
97 \( 1 + (-79.6 + 79.6i)T - 9.40e3iT^{2} \)
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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

−10.04117582943646418810825663305, −9.294818310938398221974532946401, −8.470979805753529446351183244546, −7.59426693641516896925179995557, −6.30284099574310446313206472033, −5.15386158334766888068648477477, −4.32910627972509183343305681672, −3.77139462308081546287310061277, −2.02270144832575326100124122822, −1.02749623960981454016076962827, 1.74388451049145053784064060529, 2.84926680915205027624684711104, 3.85989063604764891497627129588, 5.27896004899200632260614898174, 6.02721517451352278377021886569, 6.92100641110415920107854627847, 7.79785648245995975180977965162, 8.552925460155304347390761129673, 9.379755288704660207207310171225, 10.66492127947912950494379651034

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