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Properties

Label 1-2668-2668.107-r0-0-0
Degree $1$
Conductor $2668$
Sign $-0.993 - 0.114i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (0.591 + 0.806i)3-s + (−0.685 − 0.728i)5-s + (−0.979 + 0.202i)7-s + (−0.301 + 0.953i)9-s + (0.742 − 0.670i)11-s + (0.0203 + 0.999i)13-s + (0.182 − 0.983i)15-s + (−0.841 + 0.540i)17-s + (0.339 − 0.940i)19-s + (−0.742 − 0.670i)21-s + (−0.0611 + 0.998i)25-s + (−0.947 + 0.320i)27-s + (0.999 − 0.0407i)31-s + (0.979 + 0.202i)33-s + (0.818 + 0.574i)35-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)3-s + (−0.685 − 0.728i)5-s + (−0.979 + 0.202i)7-s + (−0.301 + 0.953i)9-s + (0.742 − 0.670i)11-s + (0.0203 + 0.999i)13-s + (0.182 − 0.983i)15-s + (−0.841 + 0.540i)17-s + (0.339 − 0.940i)19-s + (−0.742 − 0.670i)21-s + (−0.0611 + 0.998i)25-s + (−0.947 + 0.320i)27-s + (0.999 − 0.0407i)31-s + (0.979 + 0.202i)33-s + (0.818 + 0.574i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ -0.993 - 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02207207418 + 0.3829447926i\)
\(L(\frac12)\) \(\approx\) \(0.02207207418 + 0.3829447926i\)
\(L(1)\) \(\approx\) \(0.8372701146 + 0.2217779161i\)
\(L(1)\) \(\approx\) \(0.8372701146 + 0.2217779161i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.591 + 0.806i)T \)
5 \( 1 + (-0.685 - 0.728i)T \)
7 \( 1 + (-0.979 + 0.202i)T \)
11 \( 1 + (0.742 - 0.670i)T \)
13 \( 1 + (0.0203 + 0.999i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (0.339 - 0.940i)T \)
31 \( 1 + (0.999 - 0.0407i)T \)
37 \( 1 + (0.301 - 0.953i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.999 - 0.0407i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.970 - 0.242i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.882 + 0.470i)T \)
67 \( 1 + (0.742 + 0.670i)T \)
71 \( 1 + (-0.986 + 0.162i)T \)
73 \( 1 + (0.101 + 0.994i)T \)
79 \( 1 + (-0.768 + 0.639i)T \)
83 \( 1 + (-0.933 + 0.359i)T \)
89 \( 1 + (-0.182 - 0.983i)T \)
97 \( 1 + (-0.996 + 0.0815i)T \)
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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

−18.89707135259828216263225366677, −18.50016580976720616916000540228, −17.63696156534438893266879349378, −17.00965992424936572523340837290, −15.82346237455287327904563995473, −15.41023873779176701504349296754, −14.6804161669461349928538958555, −13.91150327253876564846882995842, −13.339272687043695292968397763239, −12.33901025624889217006344365197, −12.121142666834979440281636943733, −11.118544160492114495033914723090, −10.16862726072670263369447598849, −9.57910384458392177930138203542, −8.671776799848715227861748607664, −7.830565697676508251303932952545, −7.28228670203163627805054643532, −6.536409461706550702688735362284, −6.099009564005445649241002952153, −4.65461122424235831235180232597, −3.67358595003159044255300264497, −3.15673214975844117111124820522, −2.41931919806314320097437975245, −1.286446397472752772307296104430, −0.11593096053982605073781146204, 1.2957784083793271792879579232, 2.51791532471861346536015658416, 3.30367492820127695114882540814, 4.13051025354777996085892163630, 4.50568601114163530235530171553, 5.59878054027792006353012805581, 6.49627288848755762074899339563, 7.29076323991360592185634565284, 8.42803335363342193523264223706, 8.85592559166041656490594308035, 9.36141393301444039094210234206, 10.14109348717567292885693556311, 11.27062900801367880136320432583, 11.58184032797663242868878797740, 12.63241531040438224012299444450, 13.37884447548014368049779145715, 13.93288699731691229616031191540, 14.89142370325772791346665962591, 15.607538056903709458344751896761, 16.01727536789741188304714860516, 16.72700545534520283595714309724, 17.182061275284094614454065174867, 18.5431032258608935592423437663, 19.31039083101406432732618549801, 19.72705827281373367333696978248

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