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Properties

Label 2-1170-13.4-c1-0-11
Degree $2$
Conductor $1170$
Sign $0.527 - 0.849i$
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 + (1.14 − 0.661i)7-s + 0.999i·8-s + (−0.5 + 0.866i)10-s + (3.99 + 2.30i)11-s + (3.20 − 1.66i)13-s + 1.32·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (1.98 − 1.14i)19-s + (−0.866 + 0.499i)20-s + (2.30 + 3.99i)22-s + (−4.33 + 7.50i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (0.432 − 0.249i)7-s + 0.353i·8-s + (−0.158 + 0.273i)10-s + (1.20 + 0.695i)11-s + (0.887 − 0.460i)13-s + 0.353·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.455 − 0.262i)19-s + (−0.193 + 0.111i)20-s + (0.491 + 0.851i)22-s + (−0.903 + 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.527 - 0.849i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.655463558\)
\(L(\frac12)\) \(\approx\) \(2.655463558\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.20 + 1.66i)T \)
good7 \( 1 + (-1.14 + 0.661i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.99 - 2.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 - 7.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.01 - 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (-5.89 - 3.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.02 - 2.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.30 - 7.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.10iT - 47T^{2} \)
53 \( 1 + 0.826T + 53T^{2} \)
59 \( 1 + (2.72 - 1.57i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.267 + 0.464i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.75 + 1.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.81 + 5.66i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.28iT - 73T^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 + (10.2 + 5.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.63 - 4.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.675169797466206280564589728573, −9.278678826612461788018744301921, −7.85206244581797776415324947431, −7.52586224527797620437056732636, −6.41735658283344513530932775125, −5.85809208814038091954361961953, −4.61150565337369227631774114126, −3.94542360508794756095070422306, −2.86640319679756095697962114484, −1.46706815936026366543054908359, 1.12969542336146042673606271451, 2.20699698967182890680071520121, 3.73396721354986972488526500414, 4.19138868856853126226484810827, 5.38004825651482736427678585655, 6.17950062301322650712268443999, 6.85860802463442772773264128782, 8.376684859097706404942639134163, 8.664780803765445849952709122290, 9.655717121460296965295360527392

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