↑

Properties

Label 1-89-89.44-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.460 + 0.887i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.415 + 0.909i)3-s + (0.415 + 0.909i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.142 − 0.989i)7-s + (−0.142 + 0.989i)8-s + (−0.654 − 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.142 − 0.989i)11-s − 12-s + (−0.415 + 0.909i)13-s + (0.654 − 0.755i)14-s + (−0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.415 + 0.909i)3-s + (0.415 + 0.909i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.142 − 0.989i)7-s + (−0.142 + 0.989i)8-s + (−0.654 − 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.142 − 0.989i)11-s − 12-s + (−0.415 + 0.909i)13-s + (0.654 − 0.755i)14-s + (−0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6775490675 + 1.114391700i\)
\(L(\frac12)\) \(\approx\) \(0.6775490675 + 1.114391700i\)
\(L(1)\) \(\approx\) \(1.027428778 + 0.8635556888i\)
\(L(1)\) \(\approx\) \(1.027428778 + 0.8635556888i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−30.36406773451295307952885154785, −29.10395216950051751987835964989, −28.31148135350069973097539622226, −27.77447656980386587748857615329, −25.26530634021912997450594705230, −24.75219292475105464329472862719, −23.747156307392344829818547409697, −22.84288140622245566592026642754, −21.77463528653578099566425253249, −20.49908464894859699470896617123, −19.64851212579666821425474154494, −18.48781698781002783953487852898, −17.33252454404867236806309257908, −15.82552779733516088885236382342, −14.71012906130248643749916929269, −13.205539527327467140757984784895, −12.390344578519426415121196868344, −11.92271671042568847395498252796, −10.30840505050826030852069457609, −8.67846160043357548499688808874, −7.14958345921370600064647830143, −5.54887036258040251734422626075, −4.929031468123898104320893779124, −2.74766242160734832368594897052, −1.341182874449451904538033884555, 3.14059723266362222310304100897, 4.04197296316995351946527387846, 5.45354922691664856658036340609, 6.680361597653213161046712025804, 7.86157789616366950983689153387, 9.7942250289649534722889246043, 11.086852830180776512450955339, 11.80304018654021624555755752781, 13.83210827807679846598049709754, 14.35345711201385766397325890946, 15.621633981988371970521027230916, 16.55756231913834443516903743918, 17.42726106475788036653314177134, 19.08529277393541632448426734265, 20.736152620953702258493842545955, 21.4449527332921535668228808859, 22.58098662813959903838746382235, 23.173698036734093616175153753522, 24.23393895488545184784874557058, 25.831009780058115607548189702316, 26.67710721715238915659007084011, 27.17846028772480038890535616626, 29.21024433789634263323144647631, 29.76466881008778325244659668940, 31.123383012307721518396708285798

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