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Properties

Label 2-1170-13.4-c1-0-13
Degree $2$
Conductor $1170$
Sign $-0.134 + 0.990i$
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 + (−3.15 + 1.82i)7-s − 0.999i·8-s + (−0.5 + 0.866i)10-s + (1.44 + 0.834i)11-s + (−2.24 + 2.82i)13-s + 3.64·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (5.46 − 3.15i)19-s + (0.866 − 0.499i)20-s + (−0.834 − 1.44i)22-s + (−0.622 + 1.07i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (−1.19 + 0.688i)7-s − 0.353i·8-s + (−0.158 + 0.273i)10-s + (0.435 + 0.251i)11-s + (−0.622 + 0.782i)13-s + 0.974·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (1.25 − 0.724i)19-s + (0.193 − 0.111i)20-s + (−0.177 − 0.307i)22-s + (−0.129 + 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7524616409\)
\(L(\frac12)\) \(\approx\) \(0.7524616409\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 - 2.82i)T \)
good7 \( 1 + (3.15 - 1.82i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.44 - 0.834i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.46 + 3.15i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.622 - 1.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.02 + 8.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.21iT - 31T^{2} \)
37 \( 1 + (-8.54 - 4.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.04 + 4.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.78 + 6.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.82iT - 47T^{2} \)
53 \( 1 - 0.848T + 53T^{2} \)
59 \( 1 + (5.29 - 3.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.7 - 7.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.04 + 1.75i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 9.93T + 79T^{2} \)
83 \( 1 - 7.95iT - 83T^{2} \)
89 \( 1 + (5.15 + 2.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.38 - 1.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.623851239965509574765566333746, −9.046412490988261217614692628774, −8.078605030851795887713638811057, −7.02807502966216299094755885370, −6.45240536453359802314455984372, −5.27624364161932029296277901433, −4.23630950516548655963989687817, −3.02839691061777457433186930859, −2.14888817048056919264488047327, −0.45049431612109964127777416393, 1.15644177063562560081303099987, 2.88885764529529089628864547611, 3.64067014981547014206668391200, 5.01806524897364059901239182919, 6.17088990265602561046042404610, 6.67887994815967496265537929251, 7.51408432699097812551568733171, 8.281673633569295280627074591187, 9.329575028646096312554309972683, 9.983212482059529459727613500016

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