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Properties

Label 2-1170-13.10-c1-0-10
Degree $2$
Conductor $1170$
Sign $0.711 + 0.702i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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)2-s + (0.499 − 0.866i)4-s + i·5-s + (−2.59 − 1.5i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (2.59 − 1.5i)11-s + (3.5 + 0.866i)13-s + 3·14-s + (−0.5 − 0.866i)16-s + (−1.09 + 1.90i)17-s + (−5.59 − 3.23i)19-s + (0.866 + 0.499i)20-s + (−1.5 + 2.59i)22-s + (1.26 + 2.19i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−0.981 − 0.566i)7-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (0.783 − 0.452i)11-s + (0.970 + 0.240i)13-s + 0.801·14-s + (−0.125 − 0.216i)16-s + (−0.266 + 0.461i)17-s + (−1.28 − 0.741i)19-s + (0.193 + 0.111i)20-s + (−0.319 + 0.553i)22-s + (0.264 + 0.457i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9270870519\)
\(L(\frac12)\) \(\approx\) \(0.9270870519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (-3.5 - 0.866i)T \)
good7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.09 - 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.59 + 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.26 - 2.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.73 + 8.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (-9.69 + 5.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9 + 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + (9 + 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 + (-14.8 + 8.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.0 - 7.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.418870172309672345830881752242, −9.068418019524665770350618083426, −8.022908863751494342139422193664, −7.14814132730202378956134266091, −6.27090393275690705864208186506, −6.04178983512528816440843622668, −4.26874607948717822400617284335, −3.54147265847013944583948523597, −2.16139790660953337069016226905, −0.55822275020062792126481433454, 1.18881710923897909886936943145, 2.48951494358276105127215210534, 3.59218600393237613177269188992, 4.51984528597314185456564094023, 5.96516665487585841304358656757, 6.46236638507043188175731955225, 7.53591160752189231058046441444, 8.569510544966661948373505189431, 9.079033622319985794114210184787, 9.701640486519183313062804329701

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