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Properties

Label 2-690-5.3-c2-0-33
Degree $2$
Conductor $690$
Sign $-0.731 + 0.681i$
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 + (−4.34 + 2.47i)5-s + 2.44·6-s + (−5.93 + 5.93i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−1.86 + 6.82i)10-s + 11.6·11-s + (2.44 − 2.44i)12-s + (−16.3 − 16.3i)13-s + 11.8i·14-s + (−8.35 − 2.28i)15-s − 4·16-s + (2.76 − 2.76i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.868 + 0.495i)5-s + 0.408·6-s + (−0.848 + 0.848i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.186 + 0.682i)10-s + 1.06·11-s + (0.204 − 0.204i)12-s + (−1.25 − 1.25i)13-s + 0.848i·14-s + (−0.556 − 0.152i)15-s − 0.250·16-s + (0.162 − 0.162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.731 + 0.681i$
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: \((2,\ 690,\ (\ :1),\ -0.731 + 0.681i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9117036100\)
\(L(\frac12)\) \(\approx\) \(0.9117036100\)
\(L(2)\) 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 \( 1 + (4.34 - 2.47i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (5.93 - 5.93i)T - 49iT^{2} \)
11 \( 1 - 11.6T + 121T^{2} \)
13 \( 1 + (16.3 + 16.3i)T + 169iT^{2} \)
17 \( 1 + (-2.76 + 2.76i)T - 289iT^{2} \)
19 \( 1 + 34.0iT - 361T^{2} \)
29 \( 1 + 57.1iT - 841T^{2} \)
31 \( 1 + 27.9T + 961T^{2} \)
37 \( 1 + (-11.9 + 11.9i)T - 1.36e3iT^{2} \)
41 \( 1 + 66.6T + 1.68e3T^{2} \)
43 \( 1 + (-47.1 - 47.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (30.7 - 30.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (31.3 + 31.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 34.2iT - 3.48e3T^{2} \)
61 \( 1 + 59.4T + 3.72e3T^{2} \)
67 \( 1 + (14.6 - 14.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 75.9T + 5.04e3T^{2} \)
73 \( 1 + (-77.3 - 77.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 + (74.9 + 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 35.0iT - 7.92e3T^{2} \)
97 \( 1 + (-6.39 + 6.39i)T - 9.40e3iT^{2} \)
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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

−9.728719353247369685701392553829, −9.472006487213778119513744037441, −8.279823573586749214461665201623, −7.25068889573371986866497110916, −6.32400248145426179267606434530, −5.16332992884449187615904636330, −4.18000859585338313684161817696, −3.09427423974992554253228665306, −2.57671257443763120949875056001, −0.25557600418118758108555315078, 1.56945075372862992748646576775, 3.47096635258312281700892855928, 3.92829421234875267341726256506, 5.02833358041916491378080332308, 6.46654508689797460773125937207, 7.07165970017482517386085096662, 7.71467897113790605638405128279, 8.814130020279115144402152533553, 9.459777263795856942709756492620, 10.58141871715477963980594868943

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