↑

Properties

Label 2-1170-65.9-c1-0-26
Degree $2$
Conductor $1170$
Sign $0.999 + 0.00975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.67 − 1.48i)5-s + (−1.56 + 0.903i)7-s + 0.999i·8-s + (2.19 − 0.445i)10-s + (3.22 − 5.58i)11-s + (1.98 + 3.01i)13-s − 1.80·14-s + (−0.5 + 0.866i)16-s + (0.416 − 0.240i)17-s + (−3.14 − 5.44i)19-s + (2.12 + 0.710i)20-s + (5.58 − 3.22i)22-s + (6.16 + 3.55i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.749 − 0.662i)5-s + (−0.591 + 0.341i)7-s + 0.353i·8-s + (0.692 − 0.140i)10-s + (0.971 − 1.68i)11-s + (0.549 + 0.835i)13-s − 0.482·14-s + (−0.125 + 0.216i)16-s + (0.101 − 0.0583i)17-s + (−0.721 − 1.24i)19-s + (0.474 + 0.158i)20-s + (1.18 − 0.686i)22-s + (1.28 + 0.742i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.999 + 0.00975i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.999 + 0.00975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.693463137\)
\(L(\frac12)\) \(\approx\) \(2.693463137\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 1.48i)T \)
13 \( 1 + (-1.98 - 3.01i)T \)
good7 \( 1 + (1.56 - 0.903i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.22 + 5.58i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.416 + 0.240i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.16 - 3.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.15 - 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 + (-2.65 - 1.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.75 + 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 - 0.906iT - 53T^{2} \)
59 \( 1 + (3.28 + 5.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.47 - 9.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.562 - 0.324i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.83 - 3.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.60iT - 73T^{2} \)
79 \( 1 - 2.29T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + (-0.578 + 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.9 - 6.91i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.323993501889592846953116110550, −9.060598453531374084471021614729, −8.352723005319735394466887603452, −6.91283010555754377896117752794, −6.30151224055649842980903306102, −5.68364362979414437872077791606, −4.69572077280221642285657001322, −3.66162667830019186927955025217, −2.66725520716046905798863272234, −1.13018284177219097172737849359, 1.43572003858230751838165446505, 2.54183474361901471329931219955, 3.58683572696454769651684129206, 4.44954189109219354386742137450, 5.58529056168032504904259775224, 6.51448493652664556372057801343, 6.87894658417991603608810362643, 8.056423028044250580846652362948, 9.360172625346000330918728722654, 9.920090189338594125651186554185

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