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Properties

Label 2-1170-13.10-c1-0-1
Degree $2$
Conductor $1170$
Sign $-0.999 - 0.0183i$
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 + (−1.24 − 0.719i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (−3.49 + 2.01i)11-s + (3.13 − 1.78i)13-s + 1.43·14-s + (−0.5 − 0.866i)16-s + (−1.15 + 1.99i)17-s + (−0.346 − 0.199i)19-s + (−0.866 − 0.499i)20-s + (2.01 − 3.49i)22-s + (−0.780 − 1.35i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.471 − 0.272i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (−1.05 + 0.608i)11-s + (0.868 − 0.495i)13-s + 0.384·14-s + (−0.125 − 0.216i)16-s + (−0.279 + 0.484i)17-s + (−0.0794 − 0.0458i)19-s + (−0.193 − 0.111i)20-s + (0.430 − 0.745i)22-s + (−0.162 − 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.999 - 0.0183i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.999 - 0.0183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03687500328\)
\(L(\frac12)\) \(\approx\) \(0.03687500328\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.13 + 1.78i)T \)
good7 \( 1 + (1.24 + 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.49 - 2.01i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.15 - 1.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.346 + 0.199i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.780 + 1.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.93 + 6.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.10iT - 31T^{2} \)
37 \( 1 + (5.40 - 3.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.659 + 0.380i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.50 - 9.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.84iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.84 + 4.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.00 - 5.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 0.931T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-0.840 + 0.485i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.6 + 6.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.05798079356582246672412902178, −9.426958957470797227927925461314, −8.332204872732677722588278212058, −7.985801075731240600786617354533, −6.92023766988303010231252576950, −6.10874326657953401018963870777, −5.23166272392813952048219000696, −4.20040069329429066266596614755, −2.92062205822896297139152361149, −1.56294525018509141425314880919, 0.01883331932015040725184724232, 1.83438160712512085534375295665, 2.98486395523118411141830724109, 3.72725075031705035499679429649, 5.18857376685318555060431092749, 6.12564009023160983733291903018, 6.99184888202638060500197671277, 7.81234905837498976932963344734, 8.743434702272029545329991063798, 9.276814519900378759103798692380

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