Properties

Label 1-4017-4017.2846-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.912 - 0.408i$
Analytic cond. $18.6548$
Root an. cond. $18.6548$
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 + 0.0615i)2-s + (0.992 − 0.122i)4-s + (−0.213 − 0.976i)5-s + (−0.650 + 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.552 − 0.833i)11-s + (0.602 − 0.798i)14-s + (0.969 − 0.243i)16-s + (−0.816 − 0.577i)17-s + (0.0307 − 0.999i)19-s + (−0.332 − 0.943i)20-s + (0.602 + 0.798i)22-s + (−0.445 − 0.895i)23-s + (−0.908 + 0.417i)25-s + ⋯
L(s)  = 1  + (−0.998 + 0.0615i)2-s + (0.992 − 0.122i)4-s + (−0.213 − 0.976i)5-s + (−0.650 + 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.552 − 0.833i)11-s + (0.602 − 0.798i)14-s + (0.969 − 0.243i)16-s + (−0.816 − 0.577i)17-s + (0.0307 − 0.999i)19-s + (−0.332 − 0.943i)20-s + (0.602 + 0.798i)22-s + (−0.445 − 0.895i)23-s + (−0.908 + 0.417i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.912 - 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1083826112 - 0.5077750500i\)
\(L(\frac12)\) \(\approx\) \(0.1083826112 - 0.5077750500i\)
\(L(1)\) \(\approx\) \(0.5381004811 - 0.1618799201i\)
\(L(1)\) \(\approx\) \(0.5381004811 - 0.1618799201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.998 + 0.0615i)T \)
5 \( 1 + (-0.213 - 0.976i)T \)
7 \( 1 + (-0.650 + 0.759i)T \)
11 \( 1 + (-0.552 - 0.833i)T \)
17 \( 1 + (-0.816 - 0.577i)T \)
19 \( 1 + (0.0307 - 0.999i)T \)
23 \( 1 + (-0.445 - 0.895i)T \)
29 \( 1 + (0.952 + 0.303i)T \)
31 \( 1 + (-0.273 + 0.961i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (0.213 - 0.976i)T \)
43 \( 1 + (0.779 + 0.626i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.881 - 0.473i)T \)
59 \( 1 + (0.650 + 0.759i)T \)
61 \( 1 + (0.0922 - 0.995i)T \)
67 \( 1 + (0.332 - 0.943i)T \)
71 \( 1 + (0.952 - 0.303i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.739 - 0.673i)T \)
83 \( 1 + (0.332 + 0.943i)T \)
89 \( 1 + (0.602 - 0.798i)T \)
97 \( 1 + (-0.816 + 0.577i)T \)
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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

−18.80946527101048672243296757712, −17.96703752467832688291794975291, −17.6462383621782033824954779310, −16.83254764252814125837004324526, −16.078596350362535638207210764130, −15.49472228143118470342647948426, −14.92750434363725703342980248547, −14.11426178096849758702873729861, −13.22379033638969548730697104050, −12.504098848445063378147922345667, −11.70782288604165120160496719746, −10.94602550574062366206087021302, −10.40943400879350329391628175544, −9.86268513283347853072863058545, −9.31028637378366546443695903318, −8.04869262893050364974265517738, −7.71020199164485247034121646939, −6.99612957029256899807265159135, −6.3519894544067539967282628936, −5.71497611354373461642159758139, −4.186209853542779735870629024, −3.71596154483056078413351027500, −2.64526575983215614933984755171, −2.16615240485119754561817024228, −1.00830781744314459359192155920, 0.27626819773939710529283716366, 0.9174715330185322955983410697, 2.26403127949878110124235434839, 2.686269224306095073808707092639, 3.708303487795324979776348422, 4.89333821147629715113828201455, 5.48998155753327613312974820173, 6.37313006464488196415429034220, 6.94739417909436539687182284304, 8.00307578621005365410204425781, 8.57685289425115538030955823379, 9.02967113938805349424467600615, 9.63027768104692782824357788293, 10.56617020477227214937803885984, 11.19994430996828936518606466973, 11.995242914748436784348491328014, 12.51996763160596903719037977578, 13.26523652275471055093505603146, 14.06483886980331398083518039943, 15.247174399412280844138264432904, 15.679984446958104342047059801693, 16.27543965651283522767701296006, 16.605912926908925343541402717155, 17.675552489430497650577042330417, 18.11328344531087601651215180014

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