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Properties

Label 2-1183-7.2-c1-0-84
Degree $2$
Conductor $1183$
Sign $-0.353 - 0.935i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  + (0.514 − 0.890i)2-s + (−1.33 − 2.30i)3-s + (0.470 + 0.815i)4-s + (−0.745 + 1.29i)5-s − 2.74·6-s + (−1.93 − 1.80i)7-s + 3.02·8-s + (−2.04 + 3.54i)9-s + (0.767 + 1.32i)10-s + (−2.02 − 3.49i)11-s + (1.25 − 2.17i)12-s + (−2.60 + 0.798i)14-s + 3.97·15-s + (0.615 − 1.06i)16-s + (−1.84 − 3.19i)17-s + (2.10 + 3.64i)18-s + ⋯
L(s)  = 1  + (0.363 − 0.630i)2-s + (−0.768 − 1.33i)3-s + (0.235 + 0.407i)4-s + (−0.333 + 0.577i)5-s − 1.11·6-s + (−0.731 − 0.681i)7-s + 1.06·8-s + (−0.682 + 1.18i)9-s + (0.242 + 0.420i)10-s + (−0.609 − 1.05i)11-s + (0.361 − 0.626i)12-s + (−0.695 + 0.213i)14-s + 1.02·15-s + (0.153 − 0.266i)16-s + (−0.447 − 0.774i)17-s + (0.496 + 0.859i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3267183429\)
\(L(\frac12)\) \(\approx\) \(0.3267183429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.93 + 1.80i)T \)
13 \( 1 \)
good2 \( 1 + (-0.514 + 0.890i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.33 + 2.30i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.745 - 1.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.02 + 3.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.84 + 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.64 + 4.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.63 - 6.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 + (-0.838 - 1.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.51 - 6.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.409T + 41T^{2} \)
43 \( 1 + 6.43T + 43T^{2} \)
47 \( 1 + (2.56 - 4.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.790 - 1.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.82 + 6.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.64 - 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.57 + 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.71T + 71T^{2} \)
73 \( 1 + (7.59 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.376 + 0.652i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.96T + 83T^{2} \)
89 \( 1 + (-1.84 + 3.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.43T + 97T^{2} \)
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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

−9.290950413573462582208175713824, −7.912375896931458017791841008199, −7.37522905147355227095744066380, −6.88125920499055679062154464983, −5.99056077769691097871571215862, −4.91333576384383431658278608657, −3.45336460075761128533639379487, −2.94913258889658829233834560126, −1.56019601458326610109359403281, −0.13093477806716391507373570675, 2.09411751147528235831730757534, 3.81754989163270019028779323774, 4.51632560238537206284853701300, 5.32362417055045810392249968535, 5.86240673128429976640538299209, 6.70039937204365989095156170235, 7.81680327876661365913695775824, 8.803230039540184683698363370491, 9.830213643550293798479627811380, 10.17146006632811127082254604930

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