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Properties

Label 1-1183-1183.1010-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.747 + 0.664i$
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  + (0.428 + 0.903i)2-s + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (−0.200 + 0.979i)5-s + (0.799 + 0.600i)6-s + (−0.970 − 0.239i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (0.568 + 0.822i)11-s + (−0.200 + 0.979i)12-s + (0.278 + 0.960i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.987 + 0.160i)18-s + 19-s + (−0.632 − 0.774i)20-s + ⋯
L(s)  = 1  + (0.428 + 0.903i)2-s + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (−0.200 + 0.979i)5-s + (0.799 + 0.600i)6-s + (−0.970 − 0.239i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (0.568 + 0.822i)11-s + (−0.200 + 0.979i)12-s + (0.278 + 0.960i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.987 + 0.160i)18-s + 19-s + (−0.632 − 0.774i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8086410497 + 2.126797671i\)
\(L(\frac12)\) \(\approx\) \(0.8086410497 + 2.126797671i\)
\(L(1)\) \(\approx\) \(1.238764140 + 0.9938071893i\)
\(L(1)\) \(\approx\) \(1.238764140 + 0.9938071893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.428 + 0.903i)T \)
3 \( 1 + (0.885 - 0.464i)T \)
5 \( 1 + (-0.200 + 0.979i)T \)
11 \( 1 + (0.568 + 0.822i)T \)
17 \( 1 + (0.278 + 0.960i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (0.799 + 0.600i)T \)
37 \( 1 + (0.799 + 0.600i)T \)
41 \( 1 + (-0.845 - 0.534i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.987 - 0.160i)T \)
53 \( 1 + (0.692 - 0.721i)T \)
59 \( 1 + (-0.200 + 0.979i)T \)
61 \( 1 + (-0.970 + 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.0402 + 0.999i)T \)
73 \( 1 + (-0.996 + 0.0804i)T \)
79 \( 1 + (0.987 - 0.160i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.200 - 0.979i)T \)
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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

−20.7164662855553459566693793482, −20.27485819185386344176136840853, −19.750800429359380177451167205476, −18.84512424685096559446352710234, −18.262621744003436739542478128772, −16.84937298598165080038819351062, −16.247903482236407846912695522549, −15.37329088054254993580960616256, −14.51399270966308040569744334572, −13.70436217291397086565298645376, −13.35434264019689185901733475550, −12.24354587973707411783778260976, −11.63884717641334273373493696747, −10.72103457405290178241509261713, −9.61828665669132612841396442506, −9.29053174940306400705002068023, −8.44365680559163860575537821033, −7.63033046011956376611768587834, −6.09399868049574899328871836215, −5.12675180404612423588423146943, −4.43471516096299985145933994002, −3.61254896938220054272579402, −2.86020418367258692794992963549, −1.752458503164498427701389763441, −0.732840984031083582298408784633, 1.53917142567842268774356656506, 2.71567112954259878102183773284, 3.598428554264963719452550130134, 4.14173998612277337853116784843, 5.53445382194737801936995594947, 6.47118608846487922899182855430, 7.16290563699374766167898471590, 7.69504892906191466512332998065, 8.53359515608884856796081799231, 9.52161612872979834115149311346, 10.187358065630480950723941955936, 11.75314946063574203663360307995, 12.18782154582379203934594239732, 13.35133360550163487897864065321, 13.81172527891483240242478437460, 14.73209217038669874336756413595, 15.05933475988778388433262765877, 15.752667542301313494042933643256, 16.92806148869247826094238171999, 17.76562778273835030635878834876, 18.32127000452095903105551142761, 19.126425392278982282362040425351, 19.91466649742486692080508696594, 20.76706304622519249273906332504, 21.793115479496340487828767881710

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