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Properties

Label 1-76-76.51-r0-0-0
Degree $1$
Conductor $76$
Sign $0.500 + 0.865i$
Analytic cond. $0.352942$
Root an. cond. $0.352942$
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  + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(0.352942\)
Root analytic conductor: \(0.352942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ 0.500 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9321143477 + 0.5377509498i\)
\(L(\frac12)\) \(\approx\) \(0.9321143477 + 0.5377509498i\)
\(L(1)\) \(\approx\) \(1.083275425 + 0.3688009218i\)
\(L(1)\) \(\approx\) \(1.083275425 + 0.3688009218i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−30.8901519138055891328569763416, −30.4373233508719717143148088473, −29.23935169779679513595953953086, −27.695568298922221355053365464214, −26.912413378708946300270934131977, −25.72729304503838901789074113619, −24.54761346080387561436370864355, −23.74736610162254961845901738538, −22.767533698567640288126175573450, −20.92126309096255140437031165643, −19.95348668582676368390951869292, −19.43508873983814070599811507949, −17.89738983853763285580433940005, −16.8439845303249881570672611677, −15.13908527205523736984817554270, −14.47808648718023653435562816591, −12.928651348724986892617019052689, −12.137319987724391234089041851995, −10.54839155745002272706008948119, −8.92177905660928762560692715503, −7.75085632550947871117223192951, −6.994520166165967935787924530046, −4.67725889905638081289774153547, −3.397860211539442799339042130334, −1.416417976299445197701065043716, 2.504816337956395313325334771078, 3.815312071022102178552961097515, 5.181451341385042579137014281, 7.23297624546300103983140796008, 8.45893248899172949020070163461, 9.42251723957678430829461886732, 11.090750401102578106829169486778, 11.96417855533267285771427328727, 13.84664928672795585447915092540, 14.83567954737768617194656715162, 15.65364239715653496265326198759, 16.823117908440465231931191976594, 18.74642922302608491078886099671, 19.29370357029850528329819059649, 20.60861877417267711552434192811, 21.673431132654939585365563905597, 22.56965817871672882095183450524, 24.18354460655690501934664943094, 24.992326403553273788266866682520, 26.39193083713256675273892162701, 27.19427376005675875271212739948, 27.8696269985036423319439921065, 29.48356012515467132434210436332, 30.86530539091789913245977928711, 31.46334149002059888774738019902

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