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Properties

Label 1-89-89.13-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.789 - 0.613i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.415 + 0.909i)2-s + (−0.0713 + 0.997i)3-s + (−0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (−0.936 + 0.349i)6-s + (0.479 + 0.877i)7-s + (−0.959 − 0.281i)8-s + (−0.989 − 0.142i)9-s + (−0.989 + 0.142i)10-s + (0.959 − 0.281i)11-s + (−0.707 − 0.707i)12-s + (0.997 + 0.0713i)13-s + (−0.599 + 0.800i)14-s + (−0.936 − 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.0713 + 0.997i)3-s + (−0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (−0.936 + 0.349i)6-s + (0.479 + 0.877i)7-s + (−0.959 − 0.281i)8-s + (−0.989 − 0.142i)9-s + (−0.989 + 0.142i)10-s + (0.959 − 0.281i)11-s + (−0.707 − 0.707i)12-s + (0.997 + 0.0713i)13-s + (−0.599 + 0.800i)14-s + (−0.936 − 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5490150511 + 1.601332604i\)
\(L(\frac12)\) \(\approx\) \(-0.5490150511 + 1.601332604i\)
\(L(1)\) \(\approx\) \(0.4969720457 + 1.105804760i\)
\(L(1)\) \(\approx\) \(0.4969720457 + 1.105804760i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.0713 + 0.997i)T \)
5 \( 1 + (-0.281 + 0.959i)T \)
7 \( 1 + (0.479 + 0.877i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (0.997 + 0.0713i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (0.599 + 0.800i)T \)
23 \( 1 + (0.800 - 0.599i)T \)
29 \( 1 + (-0.877 + 0.479i)T \)
31 \( 1 + (0.800 + 0.599i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.997 + 0.0713i)T \)
43 \( 1 + (-0.877 - 0.479i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (0.0713 + 0.997i)T \)
61 \( 1 + (0.977 + 0.212i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.281 + 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.989 - 0.142i)T \)
83 \( 1 + (0.936 - 0.349i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−29.85432027026745549861296560119, −28.61081654593921946186182441809, −28.02003244114577897214843180181, −26.74122070063358746860939353884, −25.02052073519545931085904883850, −23.999297120595367152598884330558, −23.42791731444407622994465159415, −22.27973282411726552755402870108, −20.670814600742773312373243232953, −20.07910002320710208423494391762, −19.19470523970889657967446089551, −17.80048504153585160435456523389, −16.994372077249943005553503158046, −15.09073456704793488021798915645, −13.535197031523039076082074262106, −13.2375753096660406242956565678, −11.77065071798168604729615692424, −11.16838527184820236816389613980, −9.29985240540059182960823106589, −8.17539487502112336771915635317, −6.54814292101234644646813565019, −4.98045103503348818779757280208, −3.72802827492183433047529476889, −1.68002831628617706436847526938, −0.75273342321033861997549076981, 3.06842507170265369622541929696, 4.18863903459721936206613233382, 5.6104838292321880873857066885, 6.66624768604508511331743631947, 8.35965966222948802814991574370, 9.26966658364128462066715039868, 11.00969503203145535113687892097, 11.930476029850784325423643904844, 13.87987888471349511728699130513, 14.743587676530610099006259766936, 15.50504706304235840916105847045, 16.4710473178516307099764319233, 17.78992497259283871125883799491, 18.76149298047782497473264808502, 20.579988788915633108718461237885, 21.76991270660939164321815632930, 22.36841655450405463112703560846, 23.2509438259830939143499455670, 24.74880892379274706936068809989, 25.597619154169437462226384014046, 26.83314942452216164761681526051, 27.23287629662036324185213667617, 28.52176602520614955744325024338, 30.32928704830857490819856909946, 31.11387070201251950735748949916

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