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Properties

Label 1-1183-1183.961-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.134 - 0.990i$
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  + (0.919 − 0.391i)2-s + (−0.200 − 0.979i)3-s + (0.692 − 0.721i)4-s + (0.845 + 0.534i)5-s + (−0.568 − 0.822i)6-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.919 + 0.391i)11-s + (−0.845 − 0.534i)12-s + (0.354 − 0.935i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (−0.692 + 0.721i)18-s + (0.5 + 0.866i)19-s + (0.970 − 0.239i)20-s + ⋯
L(s)  = 1  + (0.919 − 0.391i)2-s + (−0.200 − 0.979i)3-s + (0.692 − 0.721i)4-s + (0.845 + 0.534i)5-s + (−0.568 − 0.822i)6-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.919 + 0.391i)11-s + (−0.845 − 0.534i)12-s + (0.354 − 0.935i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (−0.692 + 0.721i)18-s + (0.5 + 0.866i)19-s + (0.970 − 0.239i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.134 - 0.990i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.134 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.037857157 - 2.333153276i\)
\(L(\frac12)\) \(\approx\) \(2.037857157 - 2.333153276i\)
\(L(1)\) \(\approx\) \(1.713902546 - 1.035591080i\)
\(L(1)\) \(\approx\) \(1.713902546 - 1.035591080i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.919 - 0.391i)T \)
3 \( 1 + (-0.200 - 0.979i)T \)
5 \( 1 + (0.845 + 0.534i)T \)
11 \( 1 + (0.919 + 0.391i)T \)
17 \( 1 + (-0.632 - 0.774i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (0.996 - 0.0804i)T \)
37 \( 1 + (-0.428 + 0.903i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (-0.692 - 0.721i)T \)
53 \( 1 + (-0.632 - 0.774i)T \)
59 \( 1 + (0.0402 - 0.999i)T \)
61 \( 1 + (-0.632 + 0.774i)T \)
67 \( 1 + (-0.278 + 0.960i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (0.919 + 0.391i)T \)
79 \( 1 + (0.692 + 0.721i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.885 - 0.464i)T \)
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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

−21.46318806194944807174151801912, −21.131558310967771091664103907745, −19.9445372185000674148402824668, −19.727882518300654587192576727014, −17.7455635618003500455853516783, −17.466467599824173729075615195284, −16.60461423105262155566901545793, −16.00065173848266400390511452755, −15.28926433993030004891125428084, −14.32087154768224878436779081054, −13.89610896817043160729102710165, −12.96182473835924508156347226911, −12.1110689066161963683519966786, −11.26646054529204889987841245315, −10.582079305661069231975676363573, −9.378536979180066652886427023653, −8.92542174048391381046401176230, −7.85855121610270696424555176210, −6.44650045636521862717622128534, −6.07259969307970207339700896462, −5.09377932909584393817953371401, −4.48318338526254286928324895535, −3.59171526913145474380379277413, −2.66224656830867484353850713932, −1.402719114590060673074035198555, 1.00167177797655983456896186187, 2.01450729906276466589792780539, 2.551852266111514454974475131397, 3.67449157301454058999998589278, 4.83228808216811301857551312744, 5.74171981373721894520257752483, 6.53010770759284835298169888368, 6.87946302096456383319084181534, 8.03044493059504669726506806645, 9.34934652986057522025578680906, 10.11688858398883440655543608869, 11.02914235216120815211746386118, 11.85871129953749557428472992526, 12.33745987675919277955551342868, 13.36517580598498823946862553306, 13.92602279418008323749671943664, 14.3816532468497547330365441261, 15.31739627105807104399724225764, 16.43520762913586811207043703752, 17.296581651251850969307505094605, 18.05161226912801087965335971613, 18.81953323811614808135266341604, 19.450127705660699706460685726024, 20.41012466507918298512347812326, 20.91799900319346038301336004398

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