Properties

Label 2-1140-95.64-c1-0-9
Degree $2$
Conductor $1140$
Sign $0.999 + 0.0379i$
Analytic cond. $9.10294$
Root an. cond. $3.01710$
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  + (−0.866 + 0.5i)3-s + (2.23 − 0.133i)5-s + (0.499 − 0.866i)9-s + 5·11-s + (−1.73 − i)13-s + (−1.86 + 1.23i)15-s + (5.19 − 3i)17-s + (−3.5 + 2.59i)19-s + (−5.19 − 3i)23-s + (4.96 − 0.598i)25-s + 0.999i·27-s + (0.5 − 0.866i)29-s + 3·31-s + (−4.33 + 2.5i)33-s − 2i·37-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.998 − 0.0599i)5-s + (0.166 − 0.288i)9-s + 1.50·11-s + (−0.480 − 0.277i)13-s + (−0.481 + 0.318i)15-s + (1.26 − 0.727i)17-s + (−0.802 + 0.596i)19-s + (−1.08 − 0.625i)23-s + (0.992 − 0.119i)25-s + 0.192i·27-s + (0.0928 − 0.160i)29-s + 0.538·31-s + (−0.753 + 0.435i)33-s − 0.328i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0379i$
Analytic conductor: \(9.10294\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :1/2),\ 0.999 + 0.0379i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.791850703\)
\(L(\frac12)\) \(\approx\) \(1.791850703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-2.23 + 0.133i)T \)
19 \( 1 + (3.5 - 2.59i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 + 2i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.918884778360202969640319110554, −9.205960344900208109390639651981, −8.283773716178626608630733512745, −7.13130037148227291136469827721, −6.25082629320734685420503656861, −5.71604382527117879744619288853, −4.67692873369569731273650588750, −3.72996926623664892572391886976, −2.35826849230402189882295165130, −1.05377615483316965190485585354, 1.21732075667350323593644017219, 2.20386782995372690811850964012, 3.67420783998472470280164858567, 4.72584534851249827071511508897, 5.82695454279804881233042705615, 6.32058511204070234381946496722, 7.12499597536177351128413821857, 8.182154002303752503104950103229, 9.190084669489870023217609562883, 9.789462106597177291512341336119

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