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Properties

Label 1-89-89.7-r1-0-0
Degree $1$
Conductor $89$
Sign $0.261 + 0.965i$
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  + (−0.142 − 0.989i)2-s + (0.877 − 0.479i)3-s + (−0.959 + 0.281i)4-s + (−0.909 + 0.415i)5-s + (−0.599 − 0.800i)6-s + (−0.936 − 0.349i)7-s + (0.415 + 0.909i)8-s + (0.540 − 0.841i)9-s + (0.540 + 0.841i)10-s + (−0.415 + 0.909i)11-s + (−0.707 + 0.707i)12-s + (0.479 + 0.877i)13-s + (−0.212 + 0.977i)14-s + (−0.599 + 0.800i)15-s + (0.841 − 0.540i)16-s + (−0.989 − 0.142i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.877 − 0.479i)3-s + (−0.959 + 0.281i)4-s + (−0.909 + 0.415i)5-s + (−0.599 − 0.800i)6-s + (−0.936 − 0.349i)7-s + (0.415 + 0.909i)8-s + (0.540 − 0.841i)9-s + (0.540 + 0.841i)10-s + (−0.415 + 0.909i)11-s + (−0.707 + 0.707i)12-s + (0.479 + 0.877i)13-s + (−0.212 + 0.977i)14-s + (−0.599 + 0.800i)15-s + (0.841 − 0.540i)16-s + (−0.989 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.261 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3109127259 + 0.2378837008i\)
\(L(\frac12)\) \(\approx\) \(0.3109127259 + 0.2378837008i\)
\(L(1)\) \(\approx\) \(0.6784291324 - 0.2725679191i\)
\(L(1)\) \(\approx\) \(0.6784291324 - 0.2725679191i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (0.877 - 0.479i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (-0.936 - 0.349i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.479 + 0.877i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (0.212 + 0.977i)T \)
23 \( 1 + (-0.977 + 0.212i)T \)
29 \( 1 + (-0.349 + 0.936i)T \)
31 \( 1 + (-0.977 - 0.212i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.479 + 0.877i)T \)
43 \( 1 + (-0.349 - 0.936i)T \)
47 \( 1 + (-0.281 - 0.959i)T \)
53 \( 1 + (0.281 - 0.959i)T \)
59 \( 1 + (-0.877 - 0.479i)T \)
61 \( 1 + (-0.997 - 0.0713i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.909 + 0.415i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (-0.540 - 0.841i)T \)
83 \( 1 + (0.599 + 0.800i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−30.510989333736275713837091638284, −28.55977307400187396697019381477, −27.65268318078696282862203534875, −26.609503431178442807097267913368, −25.98588811037418543758053308065, −24.80649791624207608220853022663, −24.00580081834590216090999708359, −22.68592327950582512964369618379, −21.74865017512373341939227019904, −20.08998872933616719076211750339, −19.34653870349084470066904576780, −18.248619981428601031159355884, −16.47385593255364654975404554931, −15.753627101896409474040344416836, −15.208008271213317140526057062942, −13.62017772534262530539140914578, −12.858703004840605718267779211848, −10.7769214331834311867270945128, −9.29381439125479319815289170083, −8.506096377758708153241325446953, −7.5112497831815455814954033386, −5.86932934737004287039313882981, −4.35805950652656465265321797834, −3.17760423633864244535062895728, −0.16729157804480072773517404704, 1.89077710728346220144741992779, 3.32531101198418105674333056518, 4.186283511381597290394272594063, 6.81634876162091852009806627510, 7.945359552218968261060001362460, 9.2103119780474102909491701939, 10.295704696981648372766695043789, 11.75409679440076909251264872823, 12.75525431770410534036897183652, 13.73516943669985112518248745867, 14.951793524152059545423255007566, 16.32798172554591933496270003612, 18.2107140643722399483928655638, 18.774475765635651744047752253402, 20.00831370848357116998049023072, 20.25114740701125055552626567948, 21.88064687131659438954038030511, 23.051300775200739641286728880373, 23.817260306005224193195323407868, 25.70474687586980192662004262221, 26.293676977304498190560179067545, 27.25676195750420347360482751094, 28.61510395610342333420171354901, 29.51787673588679828812113677726, 30.63644117460093307386750545850

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