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Properties

Label 2-690-345.344-c1-0-28
Degree $2$
Conductor $690$
Sign $0.974 - 0.222i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (1.65 − 0.497i)3-s + 4-s + (−1.15 + 1.91i)5-s + (1.65 − 0.497i)6-s + 3.15·7-s + 8-s + (2.50 − 1.64i)9-s + (−1.15 + 1.91i)10-s − 2.63·11-s + (1.65 − 0.497i)12-s + 3.37i·13-s + 3.15·14-s + (−0.963 + 3.75i)15-s + 16-s + 0.331i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.957 − 0.286i)3-s + 0.5·4-s + (−0.516 + 0.856i)5-s + (0.677 − 0.202i)6-s + 1.19·7-s + 0.353·8-s + (0.835 − 0.549i)9-s + (−0.365 + 0.605i)10-s − 0.795·11-s + (0.478 − 0.143i)12-s + 0.936i·13-s + 0.844·14-s + (−0.248 + 0.968i)15-s + 0.250·16-s + 0.0803i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13471 + 0.353340i\)
\(L(\frac12)\) \(\approx\) \(3.13471 + 0.353340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.65 + 0.497i)T \)
5 \( 1 + (1.15 - 1.91i)T \)
23 \( 1 + (2.19 + 4.26i)T \)
good7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 - 0.331iT - 17T^{2} \)
19 \( 1 - 2.06iT - 19T^{2} \)
29 \( 1 + 6.73iT - 29T^{2} \)
31 \( 1 - 2.38T + 31T^{2} \)
37 \( 1 + 2.62T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 13.7iT - 53T^{2} \)
59 \( 1 - 12.0iT - 59T^{2} \)
61 \( 1 + 0.685iT - 61T^{2} \)
67 \( 1 + 0.468T + 67T^{2} \)
71 \( 1 - 5.57iT - 71T^{2} \)
73 \( 1 + 7.91iT - 73T^{2} \)
79 \( 1 + 6.81iT - 79T^{2} \)
83 \( 1 + 4.16iT - 83T^{2} \)
89 \( 1 + 7.02T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−10.60305066538827660065554159711, −9.770469499826599769324096899249, −8.309613988517122682547329490253, −7.968746024164568049840745800410, −7.05471424927942445721624374819, −6.17503658387190993098180813358, −4.69336887881298356418746357817, −3.98612897744765166788639561422, −2.78058586994975837263974359835, −1.89091374884426198184271279458, 1.54546769730316440264366808847, 2.88817799648888453868522113464, 3.97486203292114936617066434259, 4.93048441479596420155923540835, 5.41526153276201147424263446134, 7.24625092683107358736370255852, 7.942336039629045255151008891225, 8.448557220996728046539909720069, 9.486149162323367120172198110949, 10.60671543805033728636316219541

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