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Properties

Label 2-1170-65.4-c1-0-22
Degree $2$
Conductor $1170$
Sign $0.642 + 0.766i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.40 + 1.74i)5-s + (−0.763 − 1.32i)7-s + 0.999·8-s + (−0.809 − 2.08i)10-s + (1.14 + 0.658i)11-s + (2.41 − 2.67i)13-s + 1.52·14-s + (−0.5 + 0.866i)16-s + (−1.35 + 0.784i)17-s + (−4.18 + 2.41i)19-s + (2.20 + 0.341i)20-s + (−1.14 + 0.658i)22-s + (−7.31 − 4.22i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.626 + 0.779i)5-s + (−0.288 − 0.500i)7-s + 0.353·8-s + (−0.255 − 0.659i)10-s + (0.343 + 0.198i)11-s + (0.669 − 0.743i)13-s + 0.408·14-s + (−0.125 + 0.216i)16-s + (−0.329 + 0.190i)17-s + (−0.959 + 0.553i)19-s + (0.494 + 0.0763i)20-s + (−0.243 + 0.140i)22-s + (−1.52 − 0.880i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6955285606\)
\(L(\frac12)\) \(\approx\) \(0.6955285606\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (1.40 - 1.74i)T \)
13 \( 1 + (-2.41 + 2.67i)T \)
good7 \( 1 + (0.763 + 1.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.14 - 0.658i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.35 - 0.784i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.18 - 2.41i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.31 + 4.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.21 + 3.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.62iT - 31T^{2} \)
37 \( 1 + (-1.40 + 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.35 + 0.784i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.58 + 2.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 13.9iT - 53T^{2} \)
59 \( 1 + (-9.07 + 5.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.38 - 2.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.8 + 7.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.98T + 73T^{2} \)
79 \( 1 - 4.87T + 79T^{2} \)
83 \( 1 + 6.39T + 83T^{2} \)
89 \( 1 + (15.9 + 9.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.963 + 1.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.820278150071654167577394643062, −8.438315624519873671857290301296, −8.175349181212411002347501167071, −7.10577059825478872499875036049, −6.48780215797841279043954403422, −5.75314809876639147804285851824, −4.25419161838217649937897110849, −3.72231845776861707453194037118, −2.23664724297614057525126502289, −0.37298608570903825537201332649, 1.24329105937154600791541904516, 2.50400285983524367855056185268, 3.83955786539093278383009065081, 4.38212601138829136047667328822, 5.62946290745644822528479146104, 6.59235606141787119912253655420, 7.67308067907548263050325346702, 8.555073725827003929207675189777, 9.013260220376735754450295554200, 9.717467607911486532142871679019

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