Properties

Label 1-668-668.55-r0-0-0
Degree $1$
Conductor $668$
Sign $0.864 + 0.503i$
Analytic cond. $3.10217$
Root an. cond. $3.10217$
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  + (0.881 − 0.472i)3-s + (0.455 + 0.890i)5-s + (0.752 + 0.658i)7-s + (0.553 − 0.832i)9-s + (−0.776 + 0.629i)11-s + (0.999 − 0.0378i)13-s + (0.822 + 0.569i)15-s + (−0.0567 + 0.998i)17-s + (−0.672 − 0.739i)19-s + (0.974 + 0.225i)21-s + (0.280 − 0.959i)23-s + (−0.584 + 0.811i)25-s + (0.0944 − 0.995i)27-s + (0.280 + 0.959i)29-s + (0.169 + 0.985i)31-s + ⋯
L(s)  = 1  + (0.881 − 0.472i)3-s + (0.455 + 0.890i)5-s + (0.752 + 0.658i)7-s + (0.553 − 0.832i)9-s + (−0.776 + 0.629i)11-s + (0.999 − 0.0378i)13-s + (0.822 + 0.569i)15-s + (−0.0567 + 0.998i)17-s + (−0.672 − 0.739i)19-s + (0.974 + 0.225i)21-s + (0.280 − 0.959i)23-s + (−0.584 + 0.811i)25-s + (0.0944 − 0.995i)27-s + (0.280 + 0.959i)29-s + (0.169 + 0.985i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.864 + 0.503i$
Analytic conductor: \(3.10217\)
Root analytic conductor: \(3.10217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 668,\ (0:\ ),\ 0.864 + 0.503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.195312726 + 0.5927102035i\)
\(L(\frac12)\) \(\approx\) \(2.195312726 + 0.5927102035i\)
\(L(1)\) \(\approx\) \(1.610310892 + 0.1770726253i\)
\(L(1)\) \(\approx\) \(1.610310892 + 0.1770726253i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (0.881 - 0.472i)T \)
5 \( 1 + (0.455 + 0.890i)T \)
7 \( 1 + (0.752 + 0.658i)T \)
11 \( 1 + (-0.776 + 0.629i)T \)
13 \( 1 + (0.999 - 0.0378i)T \)
17 \( 1 + (-0.0567 + 0.998i)T \)
19 \( 1 + (-0.672 - 0.739i)T \)
23 \( 1 + (0.280 - 0.959i)T \)
29 \( 1 + (0.280 + 0.959i)T \)
31 \( 1 + (0.169 + 0.985i)T \)
37 \( 1 + (-0.553 - 0.832i)T \)
41 \( 1 + (0.942 - 0.334i)T \)
43 \( 1 + (-0.316 - 0.948i)T \)
47 \( 1 + (-0.929 - 0.369i)T \)
53 \( 1 + (0.965 + 0.261i)T \)
59 \( 1 + (0.0567 + 0.998i)T \)
61 \( 1 + (-0.982 - 0.188i)T \)
67 \( 1 + (-0.455 + 0.890i)T \)
71 \( 1 + (-0.644 + 0.764i)T \)
73 \( 1 + (-0.954 - 0.298i)T \)
79 \( 1 + (0.614 - 0.788i)T \)
83 \( 1 + (0.997 + 0.0756i)T \)
89 \( 1 + (0.822 - 0.569i)T \)
97 \( 1 + (-0.169 + 0.985i)T \)
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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

−22.767343761717475835950310826470, −21.27863710945851209188414773511, −21.10672761415238089434930235140, −20.5482636252197438828993300629, −19.6324980708575411451486445103, −18.67543781155896625473249072349, −17.79423107393946927881049561105, −16.75364241572764296792648882790, −16.14381831789287650609180020081, −15.31863524378090848201872882538, −14.25993787978032533044450243872, −13.41361874407468117163961590459, −13.267176660749874750173512140054, −11.6878532871771551115020249224, −10.79473356398635969240334722954, −9.91963261734168811701427993847, −9.084763616024742669502857130931, −8.137085616676459977368031557290, −7.78123114673375166800303820606, −6.16973135663244892470256693862, −5.08931606485099726683132313087, −4.36747282274591223807840351920, −3.36071378194375113142818948856, −2.12919303963350727905867991072, −1.102296953701523941213969295211, 1.58729563349742307869600004121, 2.31728689402903248853315173192, 3.165250098720071562537021476171, 4.385104629161688347204856031159, 5.65790243910203914677978393030, 6.62861287170531994168988630767, 7.4089487458646604101026395955, 8.522334317414393340757794928292, 8.92298262239559828701249004065, 10.38382321910018542645055051835, 10.80481437119791302235520447805, 12.15665586116388709311525953804, 12.95770021979309131959498054910, 13.74252870478306298355350010681, 14.735001671562296645995921016126, 15.02456682224991512603983497609, 15.97598733305454864922200903291, 17.633826642769572044000247236233, 17.95808102727606870922624198581, 18.70973937082036183193961245065, 19.43299292920804169242975411664, 20.51067177373693216933210420383, 21.24705049227507690357834483071, 21.722468400650897780020162550611, 23.02585208370290992976063562937

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