Properties

Label 1-1183-1183.447-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.994 + 0.102i$
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  + (−0.663 + 0.748i)2-s + (0.354 − 0.935i)3-s + (−0.120 − 0.992i)4-s + (0.239 − 0.970i)5-s + (0.464 + 0.885i)6-s + (0.822 + 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (−0.663 − 0.748i)11-s + (−0.970 − 0.239i)12-s + (−0.822 − 0.568i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.992 − 0.120i)18-s + i·19-s + (−0.992 − 0.120i)20-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (0.354 − 0.935i)3-s + (−0.120 − 0.992i)4-s + (0.239 − 0.970i)5-s + (0.464 + 0.885i)6-s + (0.822 + 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (−0.663 − 0.748i)11-s + (−0.970 − 0.239i)12-s + (−0.822 − 0.568i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.992 − 0.120i)18-s + i·19-s + (−0.992 − 0.120i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.994 + 0.102i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.994 + 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02605656259 - 0.5092636916i\)
\(L(\frac12)\) \(\approx\) \(0.02605656259 - 0.5092636916i\)
\(L(1)\) \(\approx\) \(0.6496009466 - 0.2553007540i\)
\(L(1)\) \(\approx\) \(0.6496009466 - 0.2553007540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.663 + 0.748i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (0.239 - 0.970i)T \)
11 \( 1 + (-0.663 - 0.748i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + iT \)
23 \( 1 - T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.464 + 0.885i)T \)
37 \( 1 + (0.464 + 0.885i)T \)
41 \( 1 + (-0.935 - 0.354i)T \)
43 \( 1 + (-0.885 - 0.464i)T \)
47 \( 1 + (-0.992 - 0.120i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (-0.239 + 0.970i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (-0.992 - 0.120i)T \)
71 \( 1 + (0.935 + 0.354i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (0.935 - 0.354i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.239 - 0.970i)T \)
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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

−21.49849930818541286434792049661, −20.899176739511215451283739375838, −19.985804658418881724333695788844, −19.52612650048495894660064798465, −18.50163241807474858855048862199, −17.95317612774144364277076291380, −17.10213682507975986998521545131, −16.33675049810610053732939663323, −15.338557183704274482845594591796, −14.84360246432327209068176572901, −13.781738406859368906861132840520, −13.069088628563367106922535222717, −11.96389415459663832732385753857, −11.07077872786451632891790878200, −10.52065349511546453369355449842, −9.84553267344799636083135917255, −9.29985364626676174916972124906, −8.126600018608660573189433947529, −7.59729431109380660351056772334, −6.452915127876418371183645806881, −5.221480942067085427329403342435, −4.185929295619545217790551118765, −3.39008683100331429101288978429, −2.572808899219365707009915022645, −1.845897825522741746725163807585, 0.24076046548218396697201268890, 1.29744182815588497712475376006, 2.11294936311151133577770494517, 3.44462831276698233648185152862, 4.90627307631711405688308890357, 5.67893139034078198960297246925, 6.31884936625191381169795827299, 7.41708514935380316484475574692, 8.184709377420451269031322998392, 8.50279640661703611816787829178, 9.56473077826128094730807591474, 10.21112797477571010334612387026, 11.55214424999653668393685017754, 12.22361669439753242519616616900, 13.38898306815122439895484530233, 13.706237320304295604977855590113, 14.56675240653430578505554681356, 15.569371436260178738689493482217, 16.434627078729763815548598762857, 16.873950654887811075106297888142, 17.86079790288028718404379532515, 18.47056637759490895023459892209, 19.033522460255726795728229831346, 19.98817397770538231202651246811, 20.51050732600481763002696259725

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