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Properties

Label 2-1183-7.4-c1-0-91
Degree $2$
Conductor $1183$
Sign $0.926 - 0.375i$
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  + (−1.13 − 1.96i)2-s + (1.66 − 2.88i)3-s + (−1.56 + 2.70i)4-s + (−0.580 − 1.00i)5-s − 7.52·6-s + (−2.63 − 0.198i)7-s + 2.54·8-s + (−4.03 − 6.98i)9-s + (−1.31 + 2.27i)10-s + (−1.22 + 2.11i)11-s + (5.19 + 8.99i)12-s + (2.59 + 5.39i)14-s − 3.86·15-s + (0.247 + 0.428i)16-s + (−0.741 + 1.28i)17-s + (−9.13 + 15.8i)18-s + ⋯
L(s)  = 1  + (−0.800 − 1.38i)2-s + (0.960 − 1.66i)3-s + (−0.780 + 1.35i)4-s + (−0.259 − 0.450i)5-s − 3.07·6-s + (−0.997 − 0.0750i)7-s + 0.898·8-s + (−1.34 − 2.32i)9-s + (−0.415 + 0.720i)10-s + (−0.368 + 0.638i)11-s + (1.49 + 2.59i)12-s + (0.693 + 1.44i)14-s − 0.998·15-s + (0.0618 + 0.107i)16-s + (−0.179 + 0.311i)17-s + (−2.15 + 3.72i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3763206148\)
\(L(\frac12)\) \(\approx\) \(0.3763206148\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.198i)T \)
13 \( 1 \)
good2 \( 1 + (1.13 + 1.96i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.66 + 2.88i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.580 + 1.00i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.22 - 2.11i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.741 - 1.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.05 - 3.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.768 - 1.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.34T + 29T^{2} \)
31 \( 1 + (-2.95 + 5.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.87 + 8.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 + (-2.07 - 3.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.92 - 3.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.440 - 0.762i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.571 + 0.990i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.104 - 0.181i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.75T + 71T^{2} \)
73 \( 1 + (0.203 - 0.352i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.11 - 1.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (-1.25 - 2.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−9.049101604130813829661850465204, −8.317312935336656435533399751527, −7.68480296006595063372038683261, −6.84313889054278189206842555140, −5.86533120398227531362919148813, −3.94904940300111134217446530823, −3.11354502837904198769783410397, −2.28661459886139604245459137391, −1.37252517513209071299488955296, −0.18928796377072930568576165517, 2.95903955293179269270243361441, 3.35137671651382750611165028974, 4.79206549022143686445268179475, 5.44440304405754171749447646317, 6.60535536719558651958042322516, 7.33645890201334743290710054929, 8.443845572921756015930312320826, 8.720877549529354768281910140822, 9.547835803321659363472283985913, 10.08684485635179030789745739925

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