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Properties

Label 1-1872-1872.605-r0-0-0
Degree $1$
Conductor $1872$
Sign $-0.241 + 0.970i$
Analytic cond. $8.69353$
Root an. cond. $8.69353$
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)5-s − i·7-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + (−0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.866 − 0.5i)47-s − 49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s − i·7-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + (−0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.866 − 0.5i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(8.69353\)
Root analytic conductor: \(8.69353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (0:\ ),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6639094549 + 0.8494240026i\)
\(L(\frac12)\) \(\approx\) \(0.6639094549 + 0.8494240026i\)
\(L(1)\) \(\approx\) \(0.9795432811 + 0.1499870920i\)
\(L(1)\) \(\approx\) \(0.9795432811 + 0.1499870920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + iT \)
67 \( 1 - T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−19.88350038662677065706447007728, −19.12427851776521686008600552761, −18.2851189595879597888834558244, −17.88197268081650461750807116709, −16.98037827499053647005617044143, −16.189375587609711444080927893759, −15.63189840875136604255304471043, −14.45187109399328961100559788131, −13.90750144678719651611079350325, −13.59601194069910088237754255925, −12.50682636287873177229219441211, −11.52014245438394820098312836536, −10.80213036852235406077566521721, −10.27114168953404975303588666695, −9.55052329311751345000405365075, −8.51630123722621058911905699310, −7.547291948383377553260045208212, −7.06823272946474689199331668285, −6.06370297351700144761076813310, −5.47043822955154990019213266254, −4.257962045661481305951353252460, −3.46801445567084677598843592247, −2.66020070517748143665513257027, −1.66642630528331000668724159716, −0.36129055746774493432144801597, 1.318475154580443594634364954825, 2.14398345883383136857420265355, 2.90083983012984545721407269493, 4.32817212200697667982531289593, 4.871005703692670725258880108591, 5.759740107988935801880910663187, 6.36684224042042579362677639242, 7.54285904258614686617355206039, 8.35484787359962191010910460003, 9.04131431064988400235960399545, 9.70213101398291502919182805838, 10.46026989496316161918050076345, 11.566849896302567291431068657320, 12.2329809495813043824077125106, 12.92052884452432578658134025189, 13.43749421288670766162684774156, 14.51353451273679948914861099402, 15.26615343831697897460533500482, 15.87298481632719912787710824035, 16.610746482721032040812452592659, 17.603976811908464617392487831223, 17.95308840495494185558070891111, 18.7172615587259667463872141191, 19.94443542794140963732351737954, 20.08909979238068438034430034588

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