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Properties

Label 1-76-76.23-r1-0-0
Degree $1$
Conductor $76$
Sign $0.612 - 0.790i$
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  + (0.939 − 0.342i)3-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)27-s + (0.766 − 0.642i)29-s + (0.5 + 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)27-s + (0.766 − 0.642i)29-s + (0.5 + 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(8.16733\)
Root analytic conductor: \(8.16733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (1:\ ),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.162254081 - 1.060049322i\)
\(L(\frac12)\) \(\approx\) \(2.162254081 - 1.060049322i\)
\(L(1)\) \(\approx\) \(1.535506289 - 0.4205128632i\)
\(L(1)\) \(\approx\) \(1.535506289 - 0.4205128632i\)

Euler product

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

−31.09382951208801327315338927966, −30.273001061240937895342186042343, −29.45866672174886599781845414207, −27.542602170797550522429449143080, −26.908716296275653466630001752997, −25.88919434023004048603011893283, −25.04399068371063421914159671714, −23.62265124314751554133201734219, −22.383843095891138824796618399565, −21.33604876173404006843683514685, −20.1993441088754309835200248900, −19.31404685778174609118850234893, −18.02378937783348735544523097188, −16.79945235424061570954983297141, −15.18338145861431527221574289353, −14.41995817010940634073803243420, −13.60777944993597314708772947858, −11.78708976597268710664095716109, −10.2902318505716349272590503959, −9.56627121529059858422612303002, −7.740541321886000050839203152342, −6.98335517220437563216060837236, −4.74152429737080920400072928341, −3.419285406029333765177131969754, −1.93966523490540649054689092479, 1.26218785707447146195524269588, 2.79120482044175750804522490959, 4.57907472978597039785653688239, 6.09010058251212573496728140512, 8.01484223638271324790435511017, 8.67749940760827325011838619165, 9.875445097004126527799884166212, 11.93057259853918412345248951071, 12.7516025197025462546337721378, 14.08386272879013386032357035981, 15.02954672039477373174432443159, 16.39671596558213521590606034028, 17.69022721714365249298411992264, 18.98801181335741295382690341332, 19.87321573648843066536438917104, 21.06153564341504958492841929189, 21.79777954772631533730209370144, 23.73623250108581322939482209510, 24.76081666766510554337783219398, 25.056282288373588472518758260168, 26.64056285429653231939841054980, 27.62779252848414824123033546461, 28.80611093908748152947717985788, 29.982758837228390534930146057680, 30.99930318690665277232700842269

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