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Properties

Label 1-1976-1976.275-r0-0-0
Degree $1$
Conductor $1976$
Sign $-0.491 + 0.870i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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 + (−0.642 + 0.766i)5-s + (0.866 + 0.5i)7-s + (0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.984 + 0.173i)15-s + (−0.766 − 0.642i)17-s + (0.342 + 0.939i)21-s + (0.173 + 0.984i)23-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (0.866 − 0.5i)31-s + (0.984 + 0.173i)33-s + (−0.939 + 0.342i)35-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.642 + 0.766i)5-s + (0.866 + 0.5i)7-s + (0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.984 + 0.173i)15-s + (−0.766 − 0.642i)17-s + (0.342 + 0.939i)21-s + (0.173 + 0.984i)23-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (0.866 − 0.5i)31-s + (0.984 + 0.173i)33-s + (−0.939 + 0.342i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $-0.491 + 0.870i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ -0.491 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049823574 + 1.797563488i\)
\(L(\frac12)\) \(\approx\) \(1.049823574 + 1.797563488i\)
\(L(1)\) \(\approx\) \(1.211251733 + 0.6584056260i\)
\(L(1)\) \(\approx\) \(1.211251733 + 0.6584056260i\)

Euler product

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

−19.80321214003879104044321833055, −19.22251188821788778184189709408, −18.33423665308780024855296547020, −17.46391172765552145846266108026, −17.0546131715840880899023022500, −16.05517813125330783696810545449, −15.04812285294421290624590529936, −14.73072489022294360201509828615, −13.84474124737616053424925917165, −13.07866154467249598995935940062, −12.45063247768099950797342462989, −11.7061220133819025574310122297, −11.03029360835848055746584646693, −9.85723926027040627355545947131, −9.01423510913319448538054866904, −8.37179786371111511202530994422, −7.83611006043929163417201041721, −7.001478980307964841072204936, −6.280792573178642655078905066854, −4.92693799631776687622932065113, −4.210693805051733148474914715545, −3.660185757645795445139162798828, −2.28783228346452947590894484789, −1.56109565612703718361443550475, −0.67337534641963102063674350386, 1.36285391841218798148545735081, 2.46859116282117016129980421430, 3.1128723232784458430762541786, 4.04568628439115059213560382162, 4.63022354632450691225650628319, 5.673370219132460448204244943220, 6.69439078877635841210648968041, 7.64356356915052326742254589976, 8.14951639189067765277561290141, 9.08132168786627946866010187229, 9.55574378022285298294334895302, 10.75631212291106095208710437483, 11.28363237690798665412135826569, 11.759617957785383763256531659331, 12.985238062343391040735479219558, 14.03896238869010234006785247853, 14.30349191088009190799269179777, 15.227703053455023279619264631515, 15.51581841451498388091165474552, 16.415364608116636148527316112264, 17.31017952540154266617725939729, 18.21623941052217364939770855846, 18.849207432614868599256435264737, 19.5963715030143595226889551901, 20.0925590678376269353724210099

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