↑

Properties

Label 1-89-89.64-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.377 + 0.926i$
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  + (−0.959 + 0.281i)2-s + (0.841 + 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (−0.654 + 0.755i)7-s + (−0.654 + 0.755i)8-s + (0.415 + 0.909i)9-s + (0.415 − 0.909i)10-s + (−0.654 − 0.755i)11-s + 12-s + (0.841 + 0.540i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.841 + 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (−0.654 + 0.755i)7-s + (−0.654 + 0.755i)8-s + (0.415 + 0.909i)9-s + (0.415 − 0.909i)10-s + (−0.654 − 0.755i)11-s + 12-s + (0.841 + 0.540i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.377 + 0.926i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.377 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3774011510 + 0.5610886567i\)
\(L(\frac12)\) \(\approx\) \(0.3774011510 + 0.5610886567i\)
\(L(1)\) \(\approx\) \(0.6373157465 + 0.3973343653i\)
\(L(1)\) \(\approx\) \(0.6373157465 + 0.3973343653i\)

Euler product

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

−30.33081056255334099185548580849, −28.91403598842815537790763909023, −28.26628807257136933203053019972, −26.79234527135878141790574796992, −26.18951840361931761857295978699, −25.12866148749165186647798068538, −24.142249927413154531267964376559, −23.01854737503147907930784653603, −21.00252166622092835611616912302, −20.124046398979693905695031049602, −19.72962510995801033912442290635, −18.472406923247011902843468458262, −17.42939425393023830558813724237, −16.042028146131906986607994510178, −15.27845907190036347394583364821, −13.19800016892314159119188824115, −12.70899842760328894443125861620, −11.13540411585855359294617032273, −9.74593691289384184387857197227, −8.65529583422746241257479676852, −7.716381576177419297501150576293, −6.69749861984009201482071458759, −4.09780017477516849613894376093, −2.70571803926395155644925926268, −0.91670079108310558077209013726, 2.41149728495937529982106873734, 3.56005599560957082379164735219, 5.815763504638164788163215109849, 7.259417383378490226653940820334, 8.41377074519325117864032158618, 9.31825064411481198872258552709, 10.57664901800027179252046628484, 11.549214930053169042929307655, 13.54205232980885812005927153342, 14.91448131808602535509176799075, 15.72823083794875131712274398155, 16.35541239971156975107630833946, 18.4278581322713591053902502147, 18.85924537482197698583304192330, 19.86559446774119970640702347673, 21.03352730142591289922078072150, 22.22227645303129488364542996080, 23.67622791884759404038703575480, 24.94064779785065378385130343234, 25.90731146338357494666332338098, 26.55028519946968936074298219701, 27.4015085294490333986587478045, 28.50838863307016328330782331857, 29.63673324621961127893425913306, 31.09604479277647275097976268474

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