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Properties

Label 1-1183-1183.716-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.234 + 0.972i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0402 − 0.999i)2-s + (0.692 − 0.721i)3-s + (−0.996 + 0.0804i)4-s + (−0.632 + 0.774i)5-s + (−0.748 − 0.663i)6-s + (0.120 + 0.992i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (−0.0402 + 0.999i)11-s + (−0.632 + 0.774i)12-s + (0.120 + 0.992i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (−0.996 + 0.0804i)18-s + (−0.5 − 0.866i)19-s + (0.568 − 0.822i)20-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (0.692 − 0.721i)3-s + (−0.996 + 0.0804i)4-s + (−0.632 + 0.774i)5-s + (−0.748 − 0.663i)6-s + (0.120 + 0.992i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (−0.0402 + 0.999i)11-s + (−0.632 + 0.774i)12-s + (0.120 + 0.992i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (−0.996 + 0.0804i)18-s + (−0.5 − 0.866i)19-s + (0.568 − 0.822i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.234 + 0.972i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.234 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07423247481 - 0.09428516631i\)
\(L(\frac12)\) \(\approx\) \(-0.07423247481 - 0.09428516631i\)
\(L(1)\) \(\approx\) \(0.6543156349 - 0.4457398427i\)
\(L(1)\) \(\approx\) \(0.6543156349 - 0.4457398427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
3 \( 1 + (0.692 - 0.721i)T \)
5 \( 1 + (-0.632 + 0.774i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (0.948 - 0.316i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (-0.748 + 0.663i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
53 \( 1 + (-0.919 - 0.391i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (-0.919 + 0.391i)T \)
67 \( 1 + (0.428 + 0.903i)T \)
71 \( 1 + (-0.970 - 0.239i)T \)
73 \( 1 + (-0.0402 + 0.999i)T \)
79 \( 1 + (-0.996 - 0.0804i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.63192697301562030752679690214, −21.230146323843773037058176634018, −20.11548308509851244380831263526, −19.39545647113892531133881466043, −18.90179829305611380737805898906, −17.654442306878966615498214828202, −16.85933121157181709500457944925, −16.199519781389758696216643339651, −15.63921959828506334215969742895, −15.07483655344782764135235371240, −14.04581515101861969063231802838, −13.538778672768229388766201063293, −12.67645772799389317592001762017, −11.57759341856380468368775706389, −10.50399627982179887382245255420, −9.67175605831173157155244365117, −8.719767503106632864772817208326, −8.35768639813815297236494953632, −7.718354067670470342208158404409, −6.476581420386521982687525442923, −5.53112432726494266525705804223, −4.671761134932382673343711063483, −3.972660915059769485057526160500, −3.23374466160554124981632095300, −1.58771887903548338567433166290, 0.04440237178314034397469038443, 1.50756802542755847672381355354, 2.53620630341185317026786762057, 2.94138346813065930792455151195, 4.138491844496549645742352496512, 4.75339422111673584287723099047, 6.502266393165040841242300261032, 7.01278667438377896469838631309, 8.15423447583487220219186530413, 8.61251833164805931666077822518, 9.74087843207499515560353220571, 10.3783171297886653538882087671, 11.476333177555494183922399538112, 11.95035367101077425980355186694, 12.84957783168030145296127795826, 13.487390098964502545506596653, 14.37536043585797609419371717599, 14.972122567887115147618392293247, 15.771992596533189222796328960, 17.3274736643707462776294643152, 17.92257194620294019505412136138, 18.53286121266556980786151919155, 19.24214783556348165148897648879, 19.974151098621171385063195367, 20.292256175958544913034743892621

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