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Properties

Label 2-1170-65.29-c1-0-2
Degree $2$
Conductor $1170$
Sign $-0.528 - 0.849i$
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 + (−2.21 + 0.311i)5-s + (3.38 + 1.95i)7-s + 0.999i·8-s + (1.76 − 1.37i)10-s + (0.533 + 0.923i)11-s + (−2.24 − 2.82i)13-s − 3.90·14-s + (−0.5 − 0.866i)16-s + (1.13 + 0.655i)17-s + (−3.29 + 5.70i)19-s + (−0.837 + 2.07i)20-s + (−0.923 − 0.533i)22-s + (1.85 − 1.07i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.990 + 0.139i)5-s + (1.27 + 0.737i)7-s + 0.353i·8-s + (0.557 − 0.435i)10-s + (0.160 + 0.278i)11-s + (−0.622 − 0.782i)13-s − 1.04·14-s + (−0.125 − 0.216i)16-s + (0.275 + 0.158i)17-s + (−0.756 + 1.30i)19-s + (−0.187 + 0.463i)20-s + (−0.196 − 0.113i)22-s + (0.387 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.528 - 0.849i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8428140628\)
\(L(\frac12)\) \(\approx\) \(0.8428140628\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.311i)T \)
13 \( 1 + (2.24 + 2.82i)T \)
good7 \( 1 + (-3.38 - 1.95i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.533 - 0.923i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.13 - 0.655i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.29 - 5.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.52 - 7.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (-4.34 + 2.51i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.97 + 5.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 - i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.0967iT - 47T^{2} \)
53 \( 1 + 1.49iT - 53T^{2} \)
59 \( 1 + (3.59 - 6.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.29 - 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.73 - 13.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 - 9.69iT - 83T^{2} \)
89 \( 1 + (2.26 + 3.91i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.68 - 2.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16633641262804753030592656904, −8.853246658145846139345540133026, −8.449062406100076856570435214954, −7.66623224516892547191287803640, −7.09701056154394691125607837431, −5.80052355926800478007662332013, −5.06243258713196034606302498251, −4.05581546385273900676799147165, −2.68952435363908000306546694975, −1.40423702950872635669952505370, 0.47863440673837986375114678595, 1.80894630860771394408752555078, 3.16156523122226389012352419014, 4.44309999801564688457372369271, 4.71514346138219075101274023927, 6.43442044149738618036639312779, 7.41268166141231034939211786585, 7.80897542708803277821814693659, 8.688252699109025845540880241199, 9.372486535611260825112630127725

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