Properties

Label 1-76-76.59-r0-0-0
Degree $1$
Conductor $76$
Sign $0.877 + 0.479i$
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  + (0.173 + 0.984i)3-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(0.352942\)
Root analytic conductor: \(0.352942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076016146 + 0.2749992670i\)
\(L(\frac12)\) \(\approx\) \(1.076016146 + 0.2749992670i\)
\(L(1)\) \(\approx\) \(1.150273095 + 0.2084733119i\)
\(L(1)\) \(\approx\) \(1.150273095 + 0.2084733119i\)

Euler product

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

−31.06001787862573541116799791335, −30.078765112540710435361904166254, −29.402728107227636146973177924351, −28.224478989024658020244608444640, −26.8441071300380982412296674538, −25.55225691451983542366471913516, −24.85284508449838205936824905915, −23.936065614017094634963723082137, −22.400732159617344843118878770906, −21.642246312713165431360647010007, −20.11521973387515681970569023656, −18.93178590217621512125307237382, −18.03093645125501760515469515237, −17.28268298094271496841590136202, −15.33004031053423485118490402325, −14.241465725521671149117472563931, −13.30113422666495848989929565143, −11.99294500514438759168170799737, −10.830180222534651718700770793125, −9.11106460481238475295526519570, −7.98652112735643248739695868680, −6.46227446645607670154925049792, −5.57962342879870492795058034048, −3.04483933000142355246253235587, −1.809773695757897265479715895966, 1.98754336232758720692578245390, 4.164559712845260850923591895614, 4.93878711143296752796417034731, 6.733607671068284798456287468658, 8.56028036744905885311187634507, 9.58833440719617169025735213393, 10.59028746253172493130418878452, 12.02625946079010582183407344363, 13.72984341672920703598221590974, 14.43911824726054518522245487085, 15.94946152006113973247379143613, 16.97634182643573144821149118289, 17.74763676681157843716230133826, 19.81147118436885358014922898960, 20.528946018207635239212263690582, 21.46390425573155905847930041083, 22.51419385335899916950795408376, 23.886249772135044449760307502942, 25.06983872491767136685746241802, 26.171186973825310211743435833308, 27.0752652939372856926146406284, 28.21059097038388249497080273645, 29.0387172083755931374586696832, 30.467910258488064474708536014643, 31.5446666894516178063099333000

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