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Properties

Label 2-1190-85.84-c1-0-33
Degree $2$
Conductor $1190$
Sign $0.927 + 0.373i$
Analytic cond. $9.50219$
Root an. cond. $3.08256$
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  − i·2-s + 3.07·3-s − 4-s + (−1.96 + 1.06i)5-s − 3.07i·6-s + 7-s + i·8-s + 6.43·9-s + (1.06 + 1.96i)10-s + 0.764i·11-s − 3.07·12-s + 3.18i·13-s − i·14-s + (−6.04 + 3.25i)15-s + 16-s + (3.16 − 2.63i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.77·3-s − 0.5·4-s + (−0.880 + 0.474i)5-s − 1.25i·6-s + 0.377·7-s + 0.353i·8-s + 2.14·9-s + (0.335 + 0.622i)10-s + 0.230i·11-s − 0.886·12-s + 0.883i·13-s − 0.267i·14-s + (−1.56 + 0.841i)15-s + 0.250·16-s + (0.768 − 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.927 + 0.373i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ 0.927 + 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.715579376\)
\(L(\frac12)\) \(\approx\) \(2.715579376\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (1.96 - 1.06i)T \)
7 \( 1 - T \)
17 \( 1 + (-3.16 + 2.63i)T \)
good3 \( 1 - 3.07T + 3T^{2} \)
11 \( 1 - 0.764iT - 11T^{2} \)
13 \( 1 - 3.18iT - 13T^{2} \)
19 \( 1 + 0.771T + 19T^{2} \)
23 \( 1 - 7.52T + 23T^{2} \)
29 \( 1 - 5.40iT - 29T^{2} \)
31 \( 1 - 1.18iT - 31T^{2} \)
37 \( 1 - 0.680T + 37T^{2} \)
41 \( 1 + 8.42iT - 41T^{2} \)
43 \( 1 + 2.40iT - 43T^{2} \)
47 \( 1 - 8.96iT - 47T^{2} \)
53 \( 1 - 8.52iT - 53T^{2} \)
59 \( 1 + 6.62T + 59T^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + 1.20iT - 67T^{2} \)
71 \( 1 + 6.72iT - 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 + 7.62iT - 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + 6.60T + 89T^{2} \)
97 \( 1 + 4.02T + 97T^{2} \)
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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

−9.401495242811148496253290937585, −9.055471047291737204308203653403, −8.183300089345432441512458260518, −7.47216485784459173665995202673, −6.85958889829995856836307551244, −4.95556767467117555297971158894, −4.12842152440727205665856530110, −3.28756229815170605543719336589, −2.63762325727555026987254014199, −1.43475727361190587597911759432, 1.18058324051435116700332197376, 2.82814829498555621928286350993, 3.64933166490288678690723571268, 4.46607636182430168423245524261, 5.46893227366754689566601226069, 6.87270399336509527507515190531, 7.71026783029716296901038052408, 8.166372642155566439991933503601, 8.630080525413332430749325011758, 9.465494600867743698876341309292

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