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Properties

Label 1-1181-1181.12-r1-0-0
Degree $1$
Conductor $1181$
Sign $0.117 + 0.993i$
Analytic cond. $126.916$
Root an. cond. $126.916$
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  + (0.996 − 0.0797i)2-s + (0.951 + 0.309i)3-s + (0.987 − 0.159i)4-s + (−0.625 − 0.779i)5-s + (0.972 + 0.232i)6-s + (−0.917 − 0.398i)7-s + (0.971 − 0.237i)8-s + (0.809 + 0.587i)9-s + (−0.686 − 0.727i)10-s + (−0.931 + 0.364i)11-s + (0.988 + 0.153i)12-s + (0.242 + 0.970i)13-s + (−0.945 − 0.324i)14-s + (−0.354 − 0.935i)15-s + (0.949 − 0.314i)16-s + (0.827 − 0.561i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0797i)2-s + (0.951 + 0.309i)3-s + (0.987 − 0.159i)4-s + (−0.625 − 0.779i)5-s + (0.972 + 0.232i)6-s + (−0.917 − 0.398i)7-s + (0.971 − 0.237i)8-s + (0.809 + 0.587i)9-s + (−0.686 − 0.727i)10-s + (−0.931 + 0.364i)11-s + (0.988 + 0.153i)12-s + (0.242 + 0.970i)13-s + (−0.945 − 0.324i)14-s + (−0.354 − 0.935i)15-s + (0.949 − 0.314i)16-s + (0.827 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1181 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1181 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1181\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(126.916\)
Root analytic conductor: \(126.916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1181} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1181,\ (1:\ ),\ 0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.740520556 + 2.435714898i\)
\(L(\frac12)\) \(\approx\) \(2.740520556 + 2.435714898i\)
\(L(1)\) \(\approx\) \(2.049538135 + 0.1983564545i\)
\(L(1)\) \(\approx\) \(2.049538135 + 0.1983564545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1181 \( 1 \)
good2 \( 1 + (0.996 - 0.0797i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.625 - 0.779i)T \)
7 \( 1 + (-0.917 - 0.398i)T \)
11 \( 1 + (-0.931 + 0.364i)T \)
13 \( 1 + (0.242 + 0.970i)T \)
17 \( 1 + (0.827 - 0.561i)T \)
19 \( 1 + (-0.211 + 0.977i)T \)
23 \( 1 + (-0.999 - 0.0425i)T \)
29 \( 1 + (-0.997 - 0.0691i)T \)
31 \( 1 + (-0.180 + 0.983i)T \)
37 \( 1 + (0.756 + 0.654i)T \)
41 \( 1 + (-0.952 + 0.303i)T \)
43 \( 1 + (0.978 - 0.206i)T \)
47 \( 1 + (-0.0159 + 0.999i)T \)
53 \( 1 + (0.0903 - 0.995i)T \)
59 \( 1 + (0.741 - 0.670i)T \)
61 \( 1 + (-0.617 - 0.786i)T \)
67 \( 1 + (0.324 + 0.945i)T \)
71 \( 1 + (-0.268 + 0.963i)T \)
73 \( 1 + (0.0744 - 0.997i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.116 + 0.993i)T \)
89 \( 1 + (-0.314 + 0.949i)T \)
97 \( 1 + (-0.731 + 0.682i)T \)
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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

−20.88044668570008711268371875854, −20.02682996581346381655641384023, −19.52786391175918777932089113772, −18.73393829178738680599928524862, −18.12288903195327631837875163769, −16.64862286832902125013141361093, −15.711500799450765610845129594, −15.34268214006029404722542251825, −14.75084221493351156520192365129, −13.77318570060210163120024079961, −13.09012617464948324063799880163, −12.58135606709656849791777489202, −11.65216179425023606483560860726, −10.608904367572352251863052561732, −10.01129123803213995889929005546, −8.68663125736550531926799013057, −7.71311590886381061846884723933, −7.37738469297363292069945018270, −6.211033850893999413776273491492, −5.65282859818817613365138861770, −4.16283006470057237043294659555, −3.42517994650353806482304261562, −2.86041497637491156381839788817, −2.141463651511279676944794314036, −0.40468260899857665549821622066, 1.27677311834491742946526704366, 2.29952717455626483792885635509, 3.38675686497956495111020571057, 3.90160651441303020852984121992, 4.69328421774149383575258829921, 5.60544173759309549670009382553, 6.813787256523900539104200220642, 7.628210539671941127431132548757, 8.25341845628965914330886703837, 9.537577065080599726552722112206, 10.03814190171605560453255683473, 11.07823185997564961956887308153, 12.18179808780983555397132792387, 12.71550634366342545662452494724, 13.447655218535760424719587015363, 14.138426893551703040321385253011, 14.92038322791126836581933506209, 15.96893698698262392345008551749, 16.10454738285337278438420034987, 16.818448146599330220898976381020, 18.6188351522948059967170122746, 19.149495710082625294381734575143, 19.95600730134136324819347430466, 20.63017183634051122624401496095, 20.92291081925493182907634651821

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