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Properties

Label 1-1183-1183.289-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.908 + 0.418i$
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  + (−0.970 − 0.239i)2-s + (−0.919 + 0.391i)3-s + (0.885 + 0.464i)4-s + (−0.996 − 0.0804i)5-s + (0.987 − 0.160i)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 + (−0.996 − 0.0804i)12-s + (0.948 − 0.316i)15-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (−0.845 + 0.534i)18-s + (−0.5 + 0.866i)19-s + (−0.845 − 0.534i)20-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (−0.919 + 0.391i)3-s + (0.885 + 0.464i)4-s + (−0.996 − 0.0804i)5-s + (0.987 − 0.160i)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 + (−0.996 − 0.0804i)12-s + (0.948 − 0.316i)15-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (−0.845 + 0.534i)18-s + (−0.5 + 0.866i)19-s + (−0.845 − 0.534i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4674172080 + 0.1025776030i\)
\(L(\frac12)\) \(\approx\) \(0.4674172080 + 0.1025776030i\)
\(L(1)\) \(\approx\) \(0.4604518181 + 0.02664478757i\)
\(L(1)\) \(\approx\) \(0.4604518181 + 0.02664478757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.970 - 0.239i)T \)
3 \( 1 + (-0.919 + 0.391i)T \)
5 \( 1 + (-0.996 - 0.0804i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.692 - 0.721i)T \)
31 \( 1 + (-0.632 - 0.774i)T \)
37 \( 1 + (-0.354 + 0.935i)T \)
41 \( 1 + (0.799 + 0.600i)T \)
43 \( 1 + (-0.632 + 0.774i)T \)
47 \( 1 + (-0.845 - 0.534i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.200 - 0.979i)T \)
67 \( 1 + (-0.845 - 0.534i)T \)
71 \( 1 + (-0.919 + 0.391i)T \)
73 \( 1 + (0.278 - 0.960i)T \)
79 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + T \)
97 \( 1 + (0.428 - 0.903i)T \)
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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

−21.21406625588163318515741070578, −19.87882687977786535004494900754, −19.45961056636967066317287432783, −18.93037490478603155586124133989, −17.963265329821565865904665611921, −17.41389426114397461650096865768, −16.56207248330044773432986936911, −16.05877551282011208941384426317, −15.26115100558860696294493097592, −14.45441069295438420959748667014, −13.16063707783844648913288421255, −12.27396896685089107849444944700, −11.53918753745619380485211770263, −10.89559413509479993008309559687, −10.46377432068025980875074748152, −8.87601967522781480875993954004, −8.63633374077576004489520449023, −7.326227992326188395700129011853, −6.94338107118490966892590755371, −6.12167387156674135061379569115, −5.11846528199030036827273651056, −4.04757796696184774020165266714, −2.79784165324445509255099890792, −1.50249000743396034157067504494, −0.55389881942827771625959975085, 0.651398142665368463709973630880, 1.77666139192803620465254113385, 3.17677290432532778576284531379, 4.11846545311020216881760549108, 4.85832190914589371528938011674, 6.28105281802513644474929762262, 6.8720753608356588462554253450, 7.69703824603305572703965713242, 8.66869869760682681337645428701, 9.51198515394873581947266991906, 10.218571041421857517642630310098, 11.20065816964227391146379625212, 11.591029834910887204427125288149, 12.30292963453933635146293940901, 13.062700103041939485452790464865, 14.85792924317845315409644617476, 15.22182639806199128850970247371, 16.17465711266578723849442735123, 16.66901763757358824622308894428, 17.40992379662658652370285287096, 18.154500653401392071246983939454, 18.91622687674472486268930311782, 19.668102082069613468811845019580, 20.45169897791631398724543593614, 21.057263374250260320789580595479

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