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Properties

Label 1-89-89.80-r0-0-0
Degree $1$
Conductor $89$
Sign $0.656 + 0.754i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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  + (−0.654 + 0.755i)2-s + (−0.989 − 0.142i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.755 − 0.654i)6-s + (−0.540 + 0.841i)7-s + (0.841 + 0.540i)8-s + (0.959 + 0.281i)9-s + (0.959 − 0.281i)10-s + (0.841 − 0.540i)11-s + i·12-s + (0.989 + 0.142i)13-s + (−0.281 − 0.959i)14-s + (0.755 + 0.654i)15-s + (−0.959 + 0.281i)16-s + (0.654 + 0.755i)17-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.989 − 0.142i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.755 − 0.654i)6-s + (−0.540 + 0.841i)7-s + (0.841 + 0.540i)8-s + (0.959 + 0.281i)9-s + (0.959 − 0.281i)10-s + (0.841 − 0.540i)11-s + i·12-s + (0.989 + 0.142i)13-s + (−0.281 − 0.959i)14-s + (0.755 + 0.654i)15-s + (−0.959 + 0.281i)16-s + (0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4158444553 + 0.1894103830i\)
\(L(\frac12)\) \(\approx\) \(0.4158444553 + 0.1894103830i\)
\(L(1)\) \(\approx\) \(0.5144323667 + 0.1472591037i\)
\(L(1)\) \(\approx\) \(0.5144323667 + 0.1472591037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.989 - 0.142i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.540 + 0.841i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (-0.281 + 0.959i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (0.540 - 0.841i)T \)
31 \( 1 + (0.281 + 0.959i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.989 - 0.142i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.142 + 0.989i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (-0.989 + 0.142i)T \)
61 \( 1 + (0.909 + 0.415i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−30.01217888754078127329781047383, −29.454520926116442819088309483730, −28.01795443156340097016638756770, −27.571360189783843521789737610553, −26.53195548342700551883368913085, −25.49792228515345307695624050276, −23.55931009633606126122163018606, −22.841386912615525302033955090195, −21.998999717603721236306952066229, −20.58578950502549706198600498434, −19.5721565557788679370469044004, −18.56483802829128814991844504316, −17.49093299774479306774697367885, −16.51780393518307985036544140828, −15.52390367879574164593921219856, −13.54124501063843880880210493092, −12.22530044814754034977579853369, −11.326199425166666237972404349041, −10.50204584435196300675773573791, −9.33736709335113411198878498983, −7.51066171171937070883225793348, −6.6585216375215555994054288618, −4.39083255029108235376156378565, −3.37567813689295228204019526845, −0.93454142619470117924016011273, 1.14690964757162530892452244003, 4.14139494603407596395670769470, 5.73202826160267484691900857108, 6.464282544046523496442809289933, 8.0498314482931620606361554175, 9.053455234082121863438634909584, 10.56039320978762682643093339673, 11.75041611270494197991207859514, 12.8078684028718107841042487254, 14.61602841855421078082112963030, 16.05208822599465299468322878401, 16.30291932546810401527341501340, 17.546860419866277577080019537884, 18.89136044179471929182023582221, 19.28845131837075334139345554624, 21.13477403428199345203121119688, 22.68689232662207073919294491474, 23.30446887207431414078268710933, 24.44027107849386427959744541547, 25.15557129548040364184343529705, 26.6627391701024570174219480508, 27.74246483334562323795446624972, 28.21796986909471296285377658511, 29.16065343811021163890642678214, 30.657245490362584482923698336011

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