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Properties

Label 1-229-229.88-r1-0-0
Degree $1$
Conductor $229$
Sign $0.717 + 0.696i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.164 − 0.986i)2-s + (−0.0825 + 0.996i)3-s + (−0.945 + 0.324i)4-s + (−0.245 − 0.969i)5-s + (0.996 − 0.0825i)6-s + (−0.837 − 0.546i)7-s + (0.475 + 0.879i)8-s + (−0.986 − 0.164i)9-s + (−0.915 + 0.401i)10-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)12-s + (−0.969 + 0.245i)13-s + (−0.401 + 0.915i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s)  = 1  + (−0.164 − 0.986i)2-s + (−0.0825 + 0.996i)3-s + (−0.945 + 0.324i)4-s + (−0.245 − 0.969i)5-s + (0.996 − 0.0825i)6-s + (−0.837 − 0.546i)7-s + (0.475 + 0.879i)8-s + (−0.986 − 0.164i)9-s + (−0.915 + 0.401i)10-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)12-s + (−0.969 + 0.245i)13-s + (−0.401 + 0.915i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.717 + 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5223471589 + 0.2120294929i\)
\(L(\frac12)\) \(\approx\) \(0.5223471589 + 0.2120294929i\)
\(L(1)\) \(\approx\) \(0.6289103499 - 0.1871238583i\)
\(L(1)\) \(\approx\) \(0.6289103499 - 0.1871238583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.164 - 0.986i)T \)
3 \( 1 + (-0.0825 + 0.996i)T \)
5 \( 1 + (-0.245 - 0.969i)T \)
7 \( 1 + (-0.837 - 0.546i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.969 + 0.245i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (0.245 - 0.969i)T \)
23 \( 1 + (-0.915 - 0.401i)T \)
29 \( 1 + (0.837 + 0.546i)T \)
31 \( 1 + (-0.735 - 0.677i)T \)
37 \( 1 + (0.789 + 0.614i)T \)
41 \( 1 + (0.164 + 0.986i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (-0.164 + 0.986i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (0.614 + 0.789i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (-0.164 + 0.986i)T \)
71 \( 1 + (0.677 + 0.735i)T \)
73 \( 1 + (-0.475 - 0.879i)T \)
79 \( 1 + (0.837 - 0.546i)T \)
83 \( 1 + (0.789 - 0.614i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.879 + 0.475i)T \)
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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

−25.56594694445483913564357968252, −25.3057481518679326400284073481, −24.36149304948461883711579826021, −23.13423229703334466683934304329, −22.689907029461604134730450018482, −21.982193459091688101739668698455, −19.845838422904437791694052786364, −19.24243799611830097597973867057, −18.311848252793540665472062974324, −17.745954778553412504994429660173, −16.59864663245976452810687065958, −15.54321770471616445240585492133, −14.5030694210493745660222064015, −13.93336923866288522419278055947, −12.5383450604204023915079157154, −11.901826830748120878958723767548, −10.16294012495320440858081153596, −9.29517186026557926503343482337, −7.85775217799128477382487263730, −7.14703751188463174843615966624, −6.38401490324555603405848868218, −5.395178264655721358979043127806, −3.61868470555832755798360156222, −2.2126553440594743986954260790, −0.2569007050706288520732651870, 0.93430210473201428834004968910, 2.859697730331047866760889437699, 3.9886781847847578008727980835, 4.63241233365627908631134409812, 6.01031842667877528570740493538, 7.96086902548944982158650258884, 9.0874905645236816577777871845, 9.635853729837267849924591380640, 10.675761428214755160538769296182, 11.70315912893784980795558907078, 12.58927126789877869584235913672, 13.649585683969786545124300859649, 14.71409957261692884981909880426, 16.22615775691746509323514122524, 16.74341288783929271674991212443, 17.57250755812687893699971827956, 19.342632205189498312934863802239, 19.77938759205148809227241957111, 20.50381008452948780631585200765, 21.78384712959349740528407817131, 22.00493529431035003598082078879, 23.26077346789762964620645870423, 24.17788437912943680729699304097, 25.747267150657738580536804999066, 26.5542647281336456852773169758

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