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Properties

Label 2-1183-7.4-c1-0-39
Degree $2$
Conductor $1183$
Sign $0.443 - 0.896i$
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  + (0.777 + 1.34i)2-s + (0.244 − 0.423i)3-s + (−0.208 + 0.361i)4-s + (−0.595 − 1.03i)5-s + 0.760·6-s + (−2.10 + 1.60i)7-s + 2.46·8-s + (1.38 + 2.39i)9-s + (0.926 − 1.60i)10-s + (1.05 − 1.83i)11-s + (0.102 + 0.176i)12-s + (−3.79 − 1.58i)14-s − 0.582·15-s + (2.33 + 4.03i)16-s + (0.453 − 0.784i)17-s + (−2.14 + 3.71i)18-s + ⋯
L(s)  = 1  + (0.549 + 0.952i)2-s + (0.141 − 0.244i)3-s + (−0.104 + 0.180i)4-s + (−0.266 − 0.461i)5-s + 0.310·6-s + (−0.795 + 0.606i)7-s + 0.870·8-s + (0.460 + 0.796i)9-s + (0.292 − 0.507i)10-s + (0.319 − 0.552i)11-s + (0.0294 + 0.0510i)12-s + (−1.01 − 0.423i)14-s − 0.150·15-s + (0.582 + 1.00i)16-s + (0.109 − 0.190i)17-s + (−0.505 + 0.876i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.443 - 0.896i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.422679555\)
\(L(\frac12)\) \(\approx\) \(2.422679555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.10 - 1.60i)T \)
13 \( 1 \)
good2 \( 1 + (-0.777 - 1.34i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.244 + 0.423i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.595 + 1.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.453 + 0.784i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.34 - 5.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.79 + 3.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.51T + 29T^{2} \)
31 \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.49 - 4.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.53T + 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 + (1.59 + 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.13 - 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.87 - 3.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (2.86 - 4.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (8.87 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.20T + 97T^{2} \)
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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

−10.03812672363387026260317802677, −8.722159991304477519653849260292, −8.188326779350736915381404210398, −7.30213037947084382964322267532, −6.49030320296202459605901133032, −5.78932040341223639429079743594, −4.95525213408729393763373886049, −4.06535255803155628758156935480, −2.78485035540348492701607993728, −1.31419221073868059339927177403, 1.04924157333087649608616258112, 2.60294050096516126781072825289, 3.45405627960306554778699978084, 4.02561259918712677504957173439, 4.94446309557233005908152603539, 6.43347808451808237585485567288, 7.10097167260249919661625884375, 7.77330203229884603687212143098, 9.313622942302346456573153905952, 9.663462627797668557186692487970

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