Properties

Label 2-1170-13.10-c1-0-14
Degree $2$
Conductor $1170$
Sign $0.331 + 0.943i$
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 + (2.40 − 1.38i)11-s + (−2.76 − 2.31i)13-s − 1.43·14-s + (−0.5 − 0.866i)16-s + (2.25 − 3.90i)17-s + (−3.75 − 2.16i)19-s + (−0.866 − 0.499i)20-s + (−1.38 + 2.40i)22-s + (−2.21 − 3.84i)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 + (0.723 − 0.417i)11-s + (−0.767 − 0.641i)13-s − 0.384·14-s + (−0.125 − 0.216i)16-s + (0.546 − 0.945i)17-s + (−0.860 − 0.496i)19-s + (−0.193 − 0.111i)20-s + (−0.295 + 0.511i)22-s + (−0.462 − 0.801i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.331 + 0.943i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043126653\)
\(L(\frac12)\) \(\approx\) \(1.043126653\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.76 + 2.31i)T \)
good7 \( 1 + (-1.24 - 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.25 + 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.75 + 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.21 + 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.93 - 6.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.16iT - 31T^{2} \)
37 \( 1 + (-0.498 + 0.287i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.65 - 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.30 + 2.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 + 9.57T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.25 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.66iT - 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + (-5.15 + 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.37 - 0.793i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.512494383114540991915086068963, −8.626195806159209729594398058945, −8.249297723599332202275597682345, −7.14899287831212393741744179215, −6.45623530106105132669217752923, −5.28833140921072752214103992841, −4.75408639410364771520206293497, −3.25840332765346804424561793925, −1.97584004347765426635928375647, −0.57158826730319751105878459803, 1.45858052427435335665348663248, 2.41890424943415374058087255284, 3.81445863831446611779286012285, 4.48026135986349802069998162380, 5.97038620110803326041213868263, 6.70494549435738954248380273330, 7.71741865798792327747240238532, 8.165858829528006850850434356349, 9.369419462965303637001699899571, 9.864636667190646493845359903042

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