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Properties

Label 2-690-5.3-c2-0-7
Degree $2$
Conductor $690$
Sign $-0.294 - 0.955i$
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 + (−0.335 − 4.98i)5-s + 2.44·6-s + (−8.75 + 8.75i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−5.32 − 4.65i)10-s − 1.73·11-s + (2.44 − 2.44i)12-s + (2.77 + 2.77i)13-s + 17.5i·14-s + (5.69 − 6.52i)15-s − 4·16-s + (−13.9 + 13.9i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.0670 − 0.997i)5-s + 0.408·6-s + (−1.25 + 1.25i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.532 − 0.465i)10-s − 0.158·11-s + (0.204 − 0.204i)12-s + (0.213 + 0.213i)13-s + 1.25i·14-s + (0.379 − 0.434i)15-s − 0.250·16-s + (−0.821 + 0.821i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.294 - 0.955i$
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: \((2,\ 690,\ (\ :1),\ -0.294 - 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9050817828\)
\(L(\frac12)\) \(\approx\) \(0.9050817828\)
\(L(2)\) 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 \( 1 + (0.335 + 4.98i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (8.75 - 8.75i)T - 49iT^{2} \)
11 \( 1 + 1.73T + 121T^{2} \)
13 \( 1 + (-2.77 - 2.77i)T + 169iT^{2} \)
17 \( 1 + (13.9 - 13.9i)T - 289iT^{2} \)
19 \( 1 + 3.58iT - 361T^{2} \)
29 \( 1 - 39.9iT - 841T^{2} \)
31 \( 1 + 29.7T + 961T^{2} \)
37 \( 1 + (36.2 - 36.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 53.7T + 1.68e3T^{2} \)
43 \( 1 + (17.3 + 17.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (41.7 - 41.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-40.3 - 40.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 88.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.38T + 3.72e3T^{2} \)
67 \( 1 + (-11.2 + 11.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 47.6T + 5.04e3T^{2} \)
73 \( 1 + (-65.5 - 65.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 13.2iT - 6.24e3T^{2} \)
83 \( 1 + (102. + 102. i)T + 6.88e3iT^{2} \)
89 \( 1 - 77.1iT - 7.92e3T^{2} \)
97 \( 1 + (21.6 - 21.6i)T - 9.40e3iT^{2} \)
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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

−10.49722504322598397835102437941, −9.461046000342186088642523381190, −9.043317913609253492490919625373, −8.332269146980522720288649145469, −6.76510870632447527767943837755, −5.78718659670318566107808859813, −5.01358552068274820000499974790, −3.90208463686314048196937969266, −2.97066251977383463146614169559, −1.78563712445220496883070496735, 0.23913153898250884782523489903, 2.47270638721495868588319146488, 3.44957679408158756741170399486, 4.16275918929565872348108485857, 5.78123906377327477253486109869, 6.72170967340547239220992336402, 7.12295520552629280842240066709, 7.889585510057212185192432166667, 9.157744927444844556650672351859, 10.02603277061874738003214376584

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