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Properties

Label 2-1170-13.10-c1-0-11
Degree $2$
Conductor $1170$
Sign $0.992 - 0.125i$
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 + (3.84 + 2.21i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (2.59 − 1.5i)11-s + (2.84 − 2.21i)13-s − 4.43·14-s + (−0.5 − 0.866i)16-s + (0.842 + 0.486i)19-s + (−0.866 − 0.499i)20-s + (−1.5 + 2.59i)22-s + (0.379 + 0.657i)23-s − 25-s + (−1.35 + 3.34i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (1.45 + 0.838i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (0.783 − 0.452i)11-s + (0.788 − 0.615i)13-s − 1.18·14-s + (−0.125 − 0.216i)16-s + (0.193 + 0.111i)19-s + (−0.193 − 0.111i)20-s + (−0.319 + 0.553i)22-s + (0.0791 + 0.137i)23-s − 0.200·25-s + (−0.265 + 0.655i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.992 - 0.125i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559724534\)
\(L(\frac12)\) \(\approx\) \(1.559724534\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.84 + 2.21i)T \)
good7 \( 1 + (-3.84 - 2.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.842 - 0.486i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.379 - 0.657i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.81 + 8.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.34 + 2.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (-4.05 - 2.34i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.68 + 9.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 6.68T + 79T^{2} \)
83 \( 1 + 9.36iT - 83T^{2} \)
89 \( 1 + (6.65 - 3.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.68 - 2.70i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.563059426008867562105157385443, −8.745327812951491688862323671252, −8.323431649905888407106158146703, −7.62501346939517280061317905574, −6.39424932582709419750389020508, −5.60008018084159351496661749090, −4.90575490707453041935154305430, −3.64967140285473318445297282910, −2.08907881717467205612800350218, −1.08435929552121931410177555943, 1.20601722267894220058446654564, 2.06493655406741470529516684068, 3.65501005400278071696416560907, 4.30715323745802523826930172262, 5.49325763865221929807534471098, 6.86876107341944414811075750891, 7.23425286405169361833810798056, 8.255268552148836317586718646947, 8.887230522181052055562922382729, 9.839672127844626565410894182633

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