↑

Properties

Label 1-3724-3724.107-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.441 - 0.897i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.733 + 0.680i)3-s + (−0.222 − 0.974i)5-s + (0.0747 − 0.997i)9-s + (−0.826 + 0.563i)11-s + (−0.0747 − 0.997i)13-s + (0.826 + 0.563i)15-s + (−0.988 − 0.149i)17-s + (0.988 − 0.149i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (0.988 + 0.149i)29-s + (−0.5 + 0.866i)31-s + (0.222 − 0.974i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)3-s + (−0.222 − 0.974i)5-s + (0.0747 − 0.997i)9-s + (−0.826 + 0.563i)11-s + (−0.0747 − 0.997i)13-s + (0.826 + 0.563i)15-s + (−0.988 − 0.149i)17-s + (0.988 − 0.149i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (0.988 + 0.149i)29-s + (−0.5 + 0.866i)31-s + (0.222 − 0.974i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6912620590 - 0.4300391067i\)
\(L(\frac12)\) \(\approx\) \(0.6912620590 - 0.4300391067i\)
\(L(1)\) \(\approx\) \(0.7224584393 - 0.03521556074i\)
\(L(1)\) \(\approx\) \(0.7224584393 - 0.03521556074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.733 + 0.680i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.988 + 0.149i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.988 + 0.149i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.70045931693689880561465418507, −18.17509928441201844913657263050, −17.48757829126305703564611535567, −16.76796301281490168573785729108, −16.056152536101294907093319539717, −15.39389904814384873087720938305, −14.60980236408118487822571719941, −13.729764720126468650093996141611, −13.34854741514813947961451219675, −12.522572843141025250148724964089, −11.61025608334509979225083196040, −11.198288833351737541663832447182, −10.686156207763338039832563981464, −9.86604731857950609154343493133, −8.85169341417337670890771926756, −7.99518969097530206664967149523, −7.36322074097900011476283080227, −6.59758955285131951730217100484, −6.24550821472052303184896837627, −5.248704501639489927917500124414, −4.51388654039509107056815544611, −3.52700800420673823390824257219, −2.50962860685195456138306351740, −2.03501198726418450317456313563, −0.739190184388362522640659349327, 0.38305055452192253947860186386, 1.26555876365595326438998494470, 2.53158466624771772957158318853, 3.41278600631005131068720510663, 4.39604668445523268729786024494, 4.976736144489066780572044907363, 5.33873878682395676564158802553, 6.351796656425547176812127227802, 7.14319345809445318721086664388, 8.09896343534623879065524454554, 8.72424937021126590511234340252, 9.53360299484859319931891018880, 10.14842474328710132829610226296, 10.90496908754059836678685430916, 11.46975800665658641901223907952, 12.45589631703485150830107294220, 12.77279650107169723142693209794, 13.442574864468979777239396617537, 14.68366652389846895342752912716, 15.260514275587707960430736961224, 15.927954744877527500214019683627, 16.26466299587672156031646320851, 17.19852549432331680582276648744, 17.74221979012650192256284230119, 18.13489374782359577454120684913

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