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Properties

Label 1-6017-6017.4348-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.590 - 0.807i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.998 + 0.0483i)2-s + (−0.809 + 0.587i)3-s + (0.995 − 0.0965i)4-s + (0.715 + 0.698i)5-s + (0.779 − 0.626i)6-s + (0.836 − 0.548i)7-s + (−0.989 + 0.144i)8-s + (0.309 − 0.951i)9-s + (−0.748 − 0.663i)10-s + (−0.748 + 0.663i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.989 − 0.144i)15-s + (0.981 − 0.192i)16-s + (−0.0724 − 0.997i)17-s + (−0.262 + 0.964i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0483i)2-s + (−0.809 + 0.587i)3-s + (0.995 − 0.0965i)4-s + (0.715 + 0.698i)5-s + (0.779 − 0.626i)6-s + (0.836 − 0.548i)7-s + (−0.989 + 0.144i)8-s + (0.309 − 0.951i)9-s + (−0.748 − 0.663i)10-s + (−0.748 + 0.663i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (−0.989 − 0.144i)15-s + (0.981 − 0.192i)16-s + (−0.0724 − 0.997i)17-s + (−0.262 + 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.590 - 0.807i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.590 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2395641932 - 0.4719707860i\)
\(L(\frac12)\) \(\approx\) \(0.2395641932 - 0.4719707860i\)
\(L(1)\) \(\approx\) \(0.6117328679 + 0.009710676950i\)
\(L(1)\) \(\approx\) \(0.6117328679 + 0.009710676950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.998 + 0.0483i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.715 + 0.698i)T \)
7 \( 1 + (0.836 - 0.548i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.0724 - 0.997i)T \)
19 \( 1 + (-0.681 - 0.732i)T \)
23 \( 1 + (0.568 - 0.822i)T \)
29 \( 1 + (0.0241 - 0.999i)T \)
31 \( 1 + (-0.906 - 0.421i)T \)
37 \( 1 + (0.0241 - 0.999i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.748 + 0.663i)T \)
47 \( 1 + (-0.943 - 0.331i)T \)
53 \( 1 + (-0.168 - 0.985i)T \)
59 \( 1 + (-0.681 + 0.732i)T \)
61 \( 1 + (0.779 - 0.626i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.906 + 0.421i)T \)
73 \( 1 + (-0.443 - 0.896i)T \)
79 \( 1 + (0.399 + 0.916i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (-0.354 + 0.935i)T \)
97 \( 1 + (-0.527 + 0.849i)T \)
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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

−17.87689860682642333843303450212, −17.30311593989528594975019180038, −16.841137485166142093943887670014, −16.38972794943523229676042854435, −15.539298711490763323311531843497, −14.7570563653298972405965400439, −14.000795488214755404111832694942, −13.08200693900984221885566011339, −12.52443090805599381104446724711, −11.888295685809244415399743426723, −11.35897910405667874789814628296, −10.594717126870862389198090319243, −10.12981672936987910496981203137, −9.021434953790700781562388326519, −8.71941085330827843620324321839, −8.02095927055316576767737929016, −7.22939665241543187190246019978, −6.43242492409604490378876588843, −5.93692705303031475965997249112, −5.24601637604738860826106338918, −4.52968908966811875657585037811, −3.3242572022304256585931810384, −2.0194882254376901934875143794, −1.6570947246702969386542077720, −1.313089764320015980064049859826, 0.217812360956921747374903170145, 1.04036885409974778506250401980, 1.99883383285249227145217920083, 2.80709707445467107315936028557, 3.61504592750896420099072448, 4.716325125541771109816576427738, 5.342721878054509604660183359316, 6.09836995043148417830534916914, 6.7606216892325703035617842374, 7.29250319417749278246863432520, 8.16094536400404681327174744531, 8.94027828478565325891647154519, 9.71105330726189489398767957074, 10.18497767231691452112518109651, 10.89973148149716428961104823851, 11.147760654951520200202704945630, 11.77733488945069361969353700705, 12.84890280063198572361029917119, 13.5004255895213913940867686584, 14.689992465060695983022699328, 14.86372284721918329586387431515, 15.59930297311191845178627700510, 16.49560110253684119610501445167, 16.869722496471010349822940199117, 17.67574223635813581902830821216

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