Properties

Label 2-690-5.2-c2-0-30
Degree $2$
Conductor $690$
Sign $-0.576 + 0.817i$
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 + (−1.86 + 4.63i)5-s − 2.44·6-s + (2.71 + 2.71i)7-s + (2 − 2i)8-s − 2.99i·9-s + (6.50 − 2.77i)10-s − 20.1·11-s + (2.44 + 2.44i)12-s + (0.746 − 0.746i)13-s − 5.43i·14-s + (3.39 + 7.96i)15-s − 4·16-s + (0.556 + 0.556i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.373 + 0.927i)5-s − 0.408·6-s + (0.387 + 0.387i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.650 − 0.277i)10-s − 1.83·11-s + (0.204 + 0.204i)12-s + (0.0574 − 0.0574i)13-s − 0.387i·14-s + (0.226 + 0.531i)15-s − 0.250·16-s + (0.0327 + 0.0327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.576 + 0.817i$
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: \((2,\ 690,\ (\ :1),\ -0.576 + 0.817i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8348574745\)
\(L(\frac12)\) \(\approx\) \(0.8348574745\)
\(L(2)\) 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 \( 1 + (1.86 - 4.63i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (-2.71 - 2.71i)T + 49iT^{2} \)
11 \( 1 + 20.1T + 121T^{2} \)
13 \( 1 + (-0.746 + 0.746i)T - 169iT^{2} \)
17 \( 1 + (-0.556 - 0.556i)T + 289iT^{2} \)
19 \( 1 + 19.9iT - 361T^{2} \)
29 \( 1 + 48.8iT - 841T^{2} \)
31 \( 1 - 50.2T + 961T^{2} \)
37 \( 1 + (16.5 + 16.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 57.2T + 1.68e3T^{2} \)
43 \( 1 + (-58.6 + 58.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (41.3 + 41.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (-28.0 + 28.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 46.2iT - 3.48e3T^{2} \)
61 \( 1 + 80.6T + 3.72e3T^{2} \)
67 \( 1 + (-3.70 - 3.70i)T + 4.48e3iT^{2} \)
71 \( 1 + 45.0T + 5.04e3T^{2} \)
73 \( 1 + (19.9 - 19.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 11.9iT - 6.24e3T^{2} \)
83 \( 1 + (-51.9 + 51.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 153. iT - 7.92e3T^{2} \)
97 \( 1 + (-109. - 109. i)T + 9.40e3iT^{2} \)
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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

−10.16650040424487586321194429399, −9.074264654082219212719437348510, −8.020700435886190695912844593836, −7.72472592543574992793706882747, −6.68184471750213383848967710280, −5.46532118720835466728786363004, −4.12802097995058319611197245614, −2.72718017639550112206012560918, −2.37609537367635102066599728560, −0.35005137329993395073595834587, 1.27689795554124951882654943392, 2.88316939940725300051917292780, 4.38575125016375627089936164912, 5.03313355526629999968447871539, 6.02346646281875646677752721064, 7.62178652422675861222598454576, 7.895264571442833469999736462233, 8.686443199390544288639058804627, 9.607174679393298767384207218070, 10.44520472031456899017308986340

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