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Properties

Label 1-91-91.89-r0-0-0
Degree $1$
Conductor $91$
Sign $-0.996 + 0.0880i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
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  + i·2-s + (0.5 + 0.866i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.866 + 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s + 17-s + (−0.866 − 0.5i)18-s + (0.866 + 0.5i)19-s + (0.866 − 0.5i)20-s + ⋯
L(s)  = 1  + i·2-s + (0.5 + 0.866i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.866 + 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s + 17-s + (−0.866 − 0.5i)18-s + (0.866 + 0.5i)19-s + (0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.996 + 0.0880i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ -0.996 + 0.0880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03534366591 + 0.8015004761i\)
\(L(\frac12)\) \(\approx\) \(0.03534366591 + 0.8015004761i\)
\(L(1)\) \(\approx\) \(0.5146496401 + 0.7301645944i\)
\(L(1)\) \(\approx\) \(0.5146496401 + 0.7301645944i\)

Euler product

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

−30.031852228354263097545777987024, −28.93309464428425284408716846229, −28.08481769660085164429932026585, −26.84794863840214476436282932946, −25.942998272852721011383716729248, −24.2832981390173677561598679832, −23.647557907875354118527569094396, −22.54832134191647510938018269101, −20.9390051152276440740848123040, −20.28456457083879106929695984123, −19.169458536675276437741676239604, −18.597836873227731760393628725855, −17.29036781266315990247215648315, −15.68415751959395053590995262340, −14.17239496356595301988903901686, −13.20313965479737200445517974300, −12.21018115108075837627270373114, −11.37913626762127934668314540320, −9.73463390853729250325097208988, −8.38311923967446934779733703033, −7.670816465220531036570365364604, −5.51228773037916365837934097102, −3.84116047672297313314076861844, −2.6242400326016899610830632864, −0.85707006923429736809035257847, 3.171958825046327983919093501546, 4.340604111619788897750284360284, 5.602787340689718365151086904640, 7.45130087646593674094957915581, 8.11702318945126956867939729681, 9.584795847647568963277784390646, 10.602682884999776519214623733591, 12.34690049686158719765604044918, 13.987308232347413512511801443843, 14.77154996481389059401943008684, 15.79260068126977932878322559547, 16.3613940462886359460704034191, 17.97852347211845975363292614880, 19.02972887038829302964919745086, 20.26529413323020925582211413369, 21.576115723414783837083464608350, 22.70983208447311265610873612814, 23.41472075251023240278001205637, 24.77245256714603274528231135316, 25.974453542472688067072425210692, 26.48275683737407710664721228947, 27.50452771758079771161703755157, 28.270512844893290488584155730015, 30.32529336354749032941807912799, 31.40710735855211932735907356414

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