Properties

Label 1-1183-1183.779-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.216 - 0.976i$
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  + (−0.692 − 0.721i)2-s + (−0.919 − 0.391i)3-s + (−0.0402 + 0.999i)4-s + (−0.428 + 0.903i)5-s + (0.354 + 0.935i)6-s + (0.748 − 0.663i)8-s + (0.692 + 0.721i)9-s + (0.948 − 0.316i)10-s + (−0.692 + 0.721i)11-s + (0.428 − 0.903i)12-s + (0.748 − 0.663i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.0402 − 0.999i)18-s + (0.5 + 0.866i)19-s + (−0.885 − 0.464i)20-s + ⋯
L(s)  = 1  + (−0.692 − 0.721i)2-s + (−0.919 − 0.391i)3-s + (−0.0402 + 0.999i)4-s + (−0.428 + 0.903i)5-s + (0.354 + 0.935i)6-s + (0.748 − 0.663i)8-s + (0.692 + 0.721i)9-s + (0.948 − 0.316i)10-s + (−0.692 + 0.721i)11-s + (0.428 − 0.903i)12-s + (0.748 − 0.663i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.0402 − 0.999i)18-s + (0.5 + 0.866i)19-s + (−0.885 − 0.464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.216 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3102196197 - 0.2489469853i\)
\(L(\frac12)\) \(\approx\) \(0.3102196197 - 0.2489469853i\)
\(L(1)\) \(\approx\) \(0.4502798986 - 0.1077948284i\)
\(L(1)\) \(\approx\) \(0.4502798986 - 0.1077948284i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.692 - 0.721i)T \)
3 \( 1 + (-0.919 - 0.391i)T \)
5 \( 1 + (-0.428 + 0.903i)T \)
11 \( 1 + (-0.692 + 0.721i)T \)
17 \( 1 + (-0.200 - 0.979i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (-0.987 - 0.160i)T \)
37 \( 1 + (0.632 - 0.774i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (-0.354 + 0.935i)T \)
47 \( 1 + (0.0402 + 0.999i)T \)
53 \( 1 + (-0.200 - 0.979i)T \)
59 \( 1 + (0.996 - 0.0804i)T \)
61 \( 1 + (-0.200 + 0.979i)T \)
67 \( 1 + (0.845 - 0.534i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (-0.692 + 0.721i)T \)
79 \( 1 + (-0.0402 - 0.999i)T \)
83 \( 1 + (-0.120 + 0.992i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.568 + 0.822i)T \)
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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

−21.5434272679252813850725589010, −20.418295867340721308265443878933, −19.85440199568114720527112909670, −18.82726950411252566646129257700, −18.196615780444823657743521740930, −17.2466955051103911070859425624, −16.84838606742465494774401089968, −16.01630980918807018751671812014, −15.585963177230998365548475550172, −14.84966194950809572648595466960, −13.4749994671589713380309026380, −12.91628993443642858292149888912, −11.64360947516096444813852124877, −11.17961385561049390442752238699, −10.24604164370041571747474270184, −9.44841505980879060434173242171, −8.667838502537143428467170395510, −7.84815655082542393110873331521, −7.005965859996200400042104263, −5.87386704522300387026326583938, −5.422359758596553613943330885404, −4.59057013991145033842859783297, −3.59114841926823736951662401986, −1.7568647920276194760406776075, −0.704522544835982704590854161556, 0.364807692724847432861099654872, 1.82116454428960060858253760704, 2.55955517498930429183065859076, 3.70464807756182131876964062447, 4.638733454753749866613654001937, 5.76485427673631644166399796671, 6.92565753057047357735699894571, 7.43360287943891251789522896764, 8.08800980068635832822820557303, 9.49554063004753435715646082596, 10.17419190363387612254292801714, 10.92280138008602228593133746695, 11.45698759285514268060644836441, 12.29575420295032526007280784630, 12.857603005493874323345611776030, 13.880237340838469451196612052950, 14.92576092465129673962729007199, 16.1546758954716168449055078585, 16.33317319592103607354243023352, 17.62403314855712741858498993520, 18.150861020101497961519467012145, 18.53929321399724466327616257600, 19.29326503017943538219842471984, 20.26911129985835046951276571388, 20.89632260759968081859521140451

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