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Properties

Label 1-3021-3021.173-r1-0-0
Degree $1$
Conductor $3021$
Sign $0.418 - 0.908i$
Analytic cond. $324.651$
Root an. cond. $324.651$
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  + (0.556 − 0.830i)2-s + (−0.379 − 0.925i)4-s + (0.944 − 0.329i)5-s + (−0.692 + 0.721i)7-s + (−0.979 − 0.200i)8-s + (0.252 − 0.967i)10-s + (0.799 − 0.600i)11-s + (0.303 + 0.952i)13-s + (0.213 + 0.977i)14-s + (−0.711 + 0.702i)16-s + (0.930 − 0.367i)17-s + (−0.663 − 0.748i)20-s + (−0.0536 − 0.998i)22-s + (0.642 − 0.766i)23-s + (0.783 − 0.621i)25-s + (0.960 + 0.278i)26-s + ⋯
L(s)  = 1  + (0.556 − 0.830i)2-s + (−0.379 − 0.925i)4-s + (0.944 − 0.329i)5-s + (−0.692 + 0.721i)7-s + (−0.979 − 0.200i)8-s + (0.252 − 0.967i)10-s + (0.799 − 0.600i)11-s + (0.303 + 0.952i)13-s + (0.213 + 0.977i)14-s + (−0.711 + 0.702i)16-s + (0.930 − 0.367i)17-s + (−0.663 − 0.748i)20-s + (−0.0536 − 0.998i)22-s + (0.642 − 0.766i)23-s + (0.783 − 0.621i)25-s + (0.960 + 0.278i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ 0.418 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.266879667 - 2.090812959i\)
\(L(\frac12)\) \(\approx\) \(3.266879667 - 2.090812959i\)
\(L(1)\) \(\approx\) \(1.465473649 - 0.7580608440i\)
\(L(1)\) \(\approx\) \(1.465473649 - 0.7580608440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.556 - 0.830i)T \)
5 \( 1 + (0.944 - 0.329i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (0.799 - 0.600i)T \)
13 \( 1 + (0.303 + 0.952i)T \)
17 \( 1 + (0.930 - 0.367i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.226 - 0.974i)T \)
31 \( 1 + (-0.600 + 0.799i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (0.291 + 0.956i)T \)
43 \( 1 + (0.0938 + 0.995i)T \)
47 \( 1 + (-0.653 - 0.757i)T \)
59 \( 1 + (-0.0134 - 0.999i)T \)
61 \( 1 + (0.989 - 0.147i)T \)
67 \( 1 + (-0.133 + 0.991i)T \)
71 \( 1 + (0.995 - 0.0938i)T \)
73 \( 1 + (-0.621 + 0.783i)T \)
79 \( 1 + (0.914 - 0.404i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.523 + 0.852i)T \)
97 \( 1 + (-0.653 + 0.757i)T \)
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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

−18.87258777085802911885931598857, −17.9902133382948441958607607287, −17.46731445602524398525127849519, −16.8565591599530609105498358336, −16.35473631060000976709568589692, −15.35033443862842015983188409644, −14.763288167820417534716091120366, −14.13771479012954300333289266972, −13.481774436823702188956368641389, −12.79896363416485670820297174783, −12.392280006730360214934724294562, −11.15757338379489811510729103762, −10.4115841065125842214541266335, −9.53415041624874832820532210559, −9.13993423943583481383791392648, −7.97267883222880583399331432054, −7.18311603225722622546261079651, −6.78167248112697257813687758262, −5.71765015083028451635573357814, −5.54965625207003341454807864731, −4.29310066728212553758282079880, −3.53611806720776362906505618144, −2.9647289353056587950731179549, −1.734978130945237273139517306330, −0.64712586223211584188597745807, 0.75865970051896989535919912727, 1.45657909402871179520031427634, 2.37553377184436853522647637740, 3.06869028360947657479319961905, 3.88921183112279423529970680956, 4.86190785592978378295917214374, 5.52875150027470391639000008502, 6.367307653097385925391327068553, 6.62035289378815905343296588665, 8.331402590790118170884191048800, 9.040273853614598780688117254859, 9.57381209310215237571813338219, 10.06253962805779697701224133389, 11.17349084919399338114616749320, 11.6995496505018323468224230303, 12.464549205112987859503061783737, 13.086302692655253298555549966919, 13.69754609473541962792316241050, 14.45044573377800440147799667560, 14.866791402597524314478493411076, 16.169302449723986500275736426356, 16.4811645064601138286057512007, 17.43494580019949240455012249925, 18.39897561967695378426752414076, 18.81493610810276998160313692931

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