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Properties

Label 1-76-76.7-r1-0-0
Degree $1$
Conductor $76$
Sign $-0.977 + 0.211i$
Analytic cond. $8.16733$
Root an. cond. $8.16733$
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)3-s + (−0.5 + 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(8.16733\)
Root analytic conductor: \(8.16733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (1:\ ),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01752593053 - 0.1640539865i\)
\(L(\frac12)\) \(\approx\) \(0.01752593053 - 0.1640539865i\)
\(L(1)\) \(\approx\) \(0.7044654399 - 0.1522540874i\)
\(L(1)\) \(\approx\) \(0.7044654399 - 0.1522540874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
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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

−31.61521664767944375999915772328, −31.14571242120998598609652371868, −29.11984710604337764867700044421, −28.49218280297833190861568541714, −27.207491688666355981305211450400, −26.37682429687118586636232301053, −25.33727098838541836669239728492, −24.09814777904529875543643579986, −22.86384921416568297837926047554, −21.72557519187364443226606572055, −20.57751367733846754172852654412, −19.82632296161231857756914367241, −18.67034205808104969224464642991, −16.65452188294457496574819737460, −16.16228342721587607669769367547, −15.08952078175208113552367537957, −13.59690192595256285060133655279, −12.5094148464740004732470535683, −10.97035817670436579553586496021, −9.5788800467567226097097136179, −8.78600284889014487122995366433, −7.299543995787600802583119062176, −5.27780204062885009587922792953, −4.15214129033572408423262913625, −2.6770527478635420309532889247, 0.0702571172676305539353107504, 2.48626842802536689974483806011, 3.51770637190867887552957505464, 5.92246605051743434104751043150, 7.16158019752255778415155275196, 8.063174924982435991636341867981, 9.72344693831655193184406107201, 11.067960010178575064774801349491, 12.601719180397152173738229483540, 13.33921016024889981124715274568, 14.837162749267407710593636037118, 15.649158186390045337469313224577, 17.434131968122452183470968441924, 18.57988319194804094539414174974, 19.349384456039528711077049851, 20.2563162676432451720506971968, 21.9496540093739727883493167696, 23.06012294622356315460127874726, 23.861503345486333069016912416223, 25.307015595357778823462559361096, 26.049711762892050243587646552997, 26.97511855967602744670523203280, 28.66446321631262261226512418285, 29.56474150393750307728901488288, 30.49561981883184603385189571573

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