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Properties

Label 2-1183-7.2-c1-0-14
Degree $2$
Conductor $1183$
Sign $-0.0836 - 0.996i$
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  + (−0.166 + 0.287i)2-s + (−0.729 − 1.26i)3-s + (0.944 + 1.63i)4-s + (−0.722 + 1.25i)5-s + 0.485·6-s + (−1.36 − 2.26i)7-s − 1.29·8-s + (0.434 − 0.752i)9-s + (−0.240 − 0.416i)10-s + (2.97 + 5.15i)11-s + (1.37 − 2.38i)12-s + (0.879 − 0.0178i)14-s + 2.11·15-s + (−1.67 + 2.90i)16-s + (−2.16 − 3.74i)17-s + (0.144 + 0.250i)18-s + ⋯
L(s)  = 1  + (−0.117 + 0.203i)2-s + (−0.421 − 0.729i)3-s + (0.472 + 0.818i)4-s + (−0.323 + 0.559i)5-s + 0.198·6-s + (−0.517 − 0.855i)7-s − 0.457·8-s + (0.144 − 0.250i)9-s + (−0.0759 − 0.131i)10-s + (0.897 + 1.55i)11-s + (0.398 − 0.689i)12-s + (0.234 − 0.00477i)14-s + 0.544·15-s + (−0.418 + 0.725i)16-s + (−0.524 − 0.909i)17-s + (0.0340 + 0.0589i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0836 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0836 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.0836 - 0.996i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.0836 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.046458134\)
\(L(\frac12)\) \(\approx\) \(1.046458134\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.36 + 2.26i)T \)
13 \( 1 \)
good2 \( 1 + (0.166 - 0.287i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.729 + 1.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.722 - 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.97 - 5.15i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.978 - 1.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.270 + 0.467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + (-3.05 - 5.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.01 - 6.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.55T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 + (3.13 - 5.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.38 - 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.425 - 0.737i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.493 - 0.854i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.76T + 71T^{2} \)
73 \( 1 + (-4.56 - 7.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0655 + 0.113i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.66T + 83T^{2} \)
89 \( 1 + (4.85 - 8.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.58T + 97T^{2} \)
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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

−9.988914467450256680459474725401, −9.164482757524126623900375165648, −8.038159978525729245253205419802, −7.09441257224925344599149092112, −6.88162134733937925445259122429, −6.44107140328070839449847654152, −4.68009072602592594316520923504, −3.79185390149568561209272577157, −2.80912803741991507169478616709, −1.38553577103939756138408292923, 0.51260562632559463837431942590, 2.03965221696585184014096007751, 3.34293106129256900707944389759, 4.43338539652762702102205417762, 5.36770993999408506376852287124, 6.10094321322156557071131909197, 6.68344507721799582135498132790, 8.299902182425729358329848026650, 8.838095303242716010851547734068, 9.610189161264558716225050941008

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