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Properties

Label 1-91-91.51-r0-0-0
Degree $1$
Conductor $91$
Sign $-0.895 + 0.444i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 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 − 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s − 20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 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 − 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2406692234 + 1.027353825i\)
\(L(\frac12)\) \(\approx\) \(0.2406692234 + 1.027353825i\)
\(L(1)\) \(\approx\) \(0.6892740425 + 0.8696353288i\)
\(L(1)\) \(\approx\) \(0.6892740425 + 0.8696353288i\)

Euler product

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

−29.95359821155699326898700858312, −28.87587084584365219385991889285, −28.40663212992003068836009613686, −27.328311928290367164153329587055, −25.3617721408890023715050553787, −24.481413760046666642117981737628, −23.54553701100354085835453292494, −22.54040831553287892119354130962, −21.53852378139294584669307510060, −20.194394279562263691822620463613, −19.58600170009699673383391177244, −18.0275131560315688871158333890, −17.48959631622873567766206120266, −15.85676752580654362299875820302, −14.08191752598918359882884953424, −13.301723871875125156053057978835, −12.27991006795978360134755593087, −11.52652814843074828191417863135, −9.97945350378578500022807239730, −8.81269816802226279938783789946, −6.94089513249744075068561452632, −5.546183130163072272736642181229, −4.55955707062691486486832355079, −2.408746523625336745580762941305, −1.14235943369505904288550557643, 3.126465871917230887181296956064, 4.31562399415332628332306714529, 5.88881209662625519593145830369, 6.453664388423262250166800394160, 8.26604835022874211540735712940, 9.63259260628939853745052325211, 10.88003517564037188798645627349, 12.136264272039222689913510497717, 13.787540614292839396131629064596, 14.61355460743488984908184914014, 15.6312526759428415199888258341, 16.72963696304421723937906675523, 17.5622342749125192929085637530, 18.744936589365168626620140426441, 20.68697621316767115062234395024, 21.875240034293247891707630205877, 22.22150726762199687555768869518, 23.32174134760079859109375940207, 24.51214007520117964645209201138, 25.7430830773732659570341871335, 26.62228430646016563505425920852, 27.27297774084136146206330523330, 28.81370118606651217755119220451, 29.93381309836366976794992871491, 31.01289158696171219183553286829

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