Properties

Label 1-1183-1183.207-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: $\chi_{1183} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.234 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292426560 + 1.017551602i\)
\(L(\frac12)\) \(\approx\) \(1.292426560 + 1.017551602i\)
\(L(1)\) \(\approx\) \(1.245394216 + 0.1396349477i\)
\(L(1)\) \(\approx\) \(1.245394216 + 0.1396349477i\)

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

−21.13766696271827489482136477200, −20.165798090814355810548551034193, −19.51065348905070844687671620594, −18.33434084973433845166366469475, −18.15871441542622932659696297156, −17.176935287417371732729081826325, −16.32075694308144040482364362844, −15.847947736813656434617394109973, −14.58148231832009566349279808370, −14.08075800240042574634115354132, −13.49696563810701494126859644164, −12.72764457665578914427719109543, −12.110938679278535266360825525882, −10.65348500827521225520348434639, −9.44783974047342061706019936522, −8.98230096974358382061738571359, −8.25748045608238844004963208010, −7.57107516927810412081977455415, −6.430041581976721776554513646350, −5.99789224692061708375704623794, −4.95548441684027949894295394738, −3.961732539086963742643476877, −2.89991336739789059209533201159, −1.64240452891103075757044939309, −0.58399313293647777656456312094, 1.66696366116123084800969822104, 2.37104249348757979550478784078, 3.10945653523705280738175785569, 4.08205650232883940972368881704, 4.81094899894435760483180658935, 5.804030583010576030302300876975, 7.08003202180086790089751426671, 8.03045186110954011571035555298, 9.184376032594060212770947766952, 9.53557787272484129255751967789, 10.365126865936453086639566740777, 10.94478249961026393164322131153, 11.815084929553138707510060109171, 12.97620451243475959546960778524, 13.63800713267267023170101893255, 14.21594691577886734247113347519, 15.12434023919155480674024452367, 15.60269927040581774621380887234, 17.10727941976507449888172801771, 17.651515965499288557090989229429, 18.43948192995566311685141220096, 19.28827173583040585845611779925, 20.091457334921766171423966151215, 20.42614010369231049775658275307, 21.62328638840134767745854296533

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