Properties

Label 1-91-91.75-r1-0-0
Degree $1$
Conductor $91$
Sign $-0.788 + 0.615i$
Analytic cond. $9.77930$
Root an. cond. $9.77930$
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  − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s − 17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯
L(s)  = 1  − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s − 17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2151064502 + 0.6247675271i\)
\(L(\frac12)\) \(\approx\) \(0.2151064502 + 0.6247675271i\)
\(L(1)\) \(\approx\) \(0.6106830440 + 0.2467690916i\)
\(L(1)\) \(\approx\) \(0.6106830440 + 0.2467690916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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

−29.7903411028914696390112763083, −28.814425833175318185028905549657, −27.41229896914112040631111329430, −26.508471382197834857480262678548, −25.77801292715841176471599129472, −24.57681994928873895665000887610, −23.8411500089441045506220783710, −22.34841564424122836982332181334, −20.8533697452057705271929894364, −19.50839151767786844489816865994, −19.16481609085305524947768662745, −18.095849141108305689651192357119, −17.104191825600582518251050309269, −15.54362338851773342831342697365, −14.64510045165914288960962188284, −13.230659678736312239943180088536, −11.6585206186284410045455488870, −10.95739216662171378006943706476, −9.23450612820734506777682287273, −8.25372555161474450555614716225, −7.08426032140180207303887873495, −6.313492854207954929178026211649, −3.41308949452995160361720356217, −2.196413538767724232037338944393, −0.37603018401740660124128768566, 1.7917563806293439042133031726, 3.63674699196909575555596071052, 5.06462843012998299638659912496, 7.04673066591743698566387820733, 8.44874597364475934417901273958, 9.10131144960979223551275717546, 10.26854005333906741390694225760, 11.474081237383580897388252781960, 12.76957648650929498925668712439, 14.69006432289200646583458275363, 15.57112391775004540222998404146, 16.5484467386195881748916048507, 17.39203731196113808700126233578, 19.006914952341293141793632956018, 20.07076434014292331174251638188, 20.51450127820592646205641320957, 21.73310739751606688027673066332, 23.280933058461404572084325506388, 24.74749995410721174733429893181, 25.389369300101720680496680052385, 26.62510725000518041433122374662, 27.44942185686683267723872022608, 28.11674725912621303718839627161, 29.11386124236655447048042739017, 30.68505635012581826850719587784

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