Properties

Label 1-91-91.31-r0-0-0
Degree $1$
Conductor $91$
Sign $0.102 + 0.994i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·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 i·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s i·20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·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 i·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + (−0.866 − 0.5i)19-s i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186646655 + 1.070475057i\)
\(L(\frac12)\) \(\approx\) \(1.186646655 + 1.070475057i\)
\(L(1)\) \(\approx\) \(1.386500283 + 0.8004514847i\)
\(L(1)\) \(\approx\) \(1.386500283 + 0.8004514847i\)

Euler product

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

−30.35755777662990623653154855528, −29.57687917801926278493397107326, −28.31448627272631787550000658775, −27.15017020464785967115465152473, −25.680295256177111027691508812736, −24.711856094155098627655410128034, −23.583936471620189054134545743805, −23.01251471775169963769261993608, −21.76020524627156936275212081867, −20.38001137745892638765683700496, −19.46604333081703175783603945960, −18.95582508355419156809775500793, −17.41008259552527091132407139836, −15.46897405303767845580709367181, −14.73751721028117985272790432325, −13.70919453520482245697286401711, −12.43339657546073263090943016272, −11.76386643419881705883653609254, −10.42051852218292050611454279535, −8.6705989170105587688866177667, −7.16019062729679537476339411198, −6.29081934596235963518770365973, −4.26244891129068368514472235812, −3.15240908719239836395290315457, −1.66054099299582962652779265541, 2.865654629979179506071405545304, 4.13085678887864485888401244816, 4.903296030629353602161364968154, 6.65320349361982106561728885631, 8.17701072043816006936068967808, 9.00345964372971763786718094654, 10.960666654135417337239991725263, 11.95322001781591803122267186630, 13.35748962931002639807818663860, 14.47168352857089153890151563305, 15.433131684381002519541949808332, 16.253928984085544196941456228177, 17.14856201397698879808536663690, 19.324673479381573916714882625439, 20.24859969283365557819648267119, 21.18837914738047852327070651304, 22.25797777801824764809683667888, 23.15023497438974146180936247209, 24.442502926629565887035859093413, 25.173829359093597987276025108079, 26.546165757271908762938835375008, 27.20108441685295784886852686259, 28.44116617219413261987936746718, 30.02416244237860143471508974444, 31.022858997988430083116277525955

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