Properties

Label 2-1170-65.49-c1-0-11
Degree $2$
Conductor $1170$
Sign $0.967 + 0.254i$
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 + (−2.10 + 0.767i)5-s + (0.823 − 1.42i)7-s + 0.999·8-s + (1.71 + 1.43i)10-s + (−2.08 + 1.20i)11-s + (−3.59 − 0.256i)13-s − 1.64·14-s + (−0.5 − 0.866i)16-s + (−0.210 − 0.121i)17-s + (3.82 + 2.20i)19-s + (0.385 − 2.20i)20-s + (2.08 + 1.20i)22-s + (7.46 − 4.31i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.939 + 0.343i)5-s + (0.311 − 0.538i)7-s + 0.353·8-s + (0.542 + 0.453i)10-s + (−0.628 + 0.362i)11-s + (−0.997 − 0.0710i)13-s − 0.439·14-s + (−0.125 − 0.216i)16-s + (−0.0511 − 0.0295i)17-s + (0.877 + 0.506i)19-s + (0.0861 − 0.492i)20-s + (0.444 + 0.256i)22-s + (1.55 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9801004119\)
\(L(\frac12)\) \(\approx\) \(0.9801004119\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 0.767i)T \)
13 \( 1 + (3.59 + 0.256i)T \)
good7 \( 1 + (-0.823 + 1.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.210 + 0.121i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.82 - 2.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.46 + 4.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0221 + 0.0383i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + (-4.47 - 7.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.210 - 0.121i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.82 + 3.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 - 2.44iT - 53T^{2} \)
59 \( 1 + (-8.35 - 4.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.937 - 1.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.53 - 3.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 - 6.79T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 + (-8.69 + 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.25 + 14.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.992644466402484186504655398273, −8.935928518919712445547249137267, −8.055160959108486513211431534437, −7.42089318536742812640623485305, −6.80285858363259092688853676710, −5.12955783916437104349970716830, −4.49587182145587707848590423644, −3.36422041838138340684672794974, −2.52140495678339852271842149963, −0.870334850968159669680348967219, 0.70408615940391018632584839452, 2.48769828230872838315769937020, 3.73054984427531087639604943651, 5.10554331026410560619962803000, 5.24827093067618312696789833111, 6.69707700084139833716292590172, 7.58356077007524557947730605289, 7.948372826797698136277953452422, 9.042262225715370975134152524010, 9.420699568258027531482783999986

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