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Properties

Label 1-4017-4017.2534-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.972 + 0.232i$
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  + (−0.908 − 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.969 + 0.243i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (−0.816 + 0.577i)11-s + (0.982 + 0.183i)14-s + (−0.153 + 0.988i)16-s + (0.389 − 0.920i)17-s + (−0.213 − 0.976i)19-s + (0.696 + 0.717i)20-s + (0.982 − 0.183i)22-s + (−0.0922 + 0.995i)23-s + (0.992 − 0.122i)25-s + ⋯
L(s)  = 1  + (−0.908 − 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.969 + 0.243i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (−0.816 + 0.577i)11-s + (0.982 + 0.183i)14-s + (−0.153 + 0.988i)16-s + (0.389 − 0.920i)17-s + (−0.213 − 0.976i)19-s + (0.696 + 0.717i)20-s + (0.982 − 0.183i)22-s + (−0.0922 + 0.995i)23-s + (0.992 − 0.122i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2534, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008205301646 - 0.06963801865i\)
\(L(\frac12)\) \(\approx\) \(0.008205301646 - 0.06963801865i\)
\(L(1)\) \(\approx\) \(0.6172318637 - 0.09537884210i\)
\(L(1)\) \(\approx\) \(0.6172318637 - 0.09537884210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.908 - 0.417i)T \)
5 \( 1 + (0.998 - 0.0615i)T \)
7 \( 1 + (-0.969 + 0.243i)T \)
11 \( 1 + (-0.816 + 0.577i)T \)
17 \( 1 + (0.389 - 0.920i)T \)
19 \( 1 + (-0.213 - 0.976i)T \)
23 \( 1 + (-0.0922 + 0.995i)T \)
29 \( 1 + (-0.552 - 0.833i)T \)
31 \( 1 + (0.932 - 0.361i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (-0.998 - 0.0615i)T \)
43 \( 1 + (0.0307 + 0.999i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.952 - 0.303i)T \)
59 \( 1 + (0.969 + 0.243i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (-0.696 + 0.717i)T \)
71 \( 1 + (-0.552 + 0.833i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (-0.696 - 0.717i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (0.389 + 0.920i)T \)
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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

−18.7557565873196576993188196037, −18.26143311745038653177880232359, −17.384006485430729969776626410555, −16.659136586186542435087570617269, −16.53134823481609911125412670482, −15.555415204783278889301625806337, −14.91571062161748971895460792729, −14.05709300939598364404772632294, −13.55683721842577516719471309548, −12.669491327998857990475213352952, −12.057738494351190466478325846579, −10.73209430723325378442939534133, −10.35568978891821576422082119840, −10.054649893283981533132398619, −9.0350455844357994813653265601, −8.53423931002688958889051400475, −7.760711296643263324879410259003, −6.7856977917272431346521590108, −6.31805137570141462699879907452, −5.6781790935852877734520694480, −5.0079041678707577094614826002, −3.60447484860832719136339713653, −2.850697827043810449023554213359, −1.98955736623555507140617666459, −1.18285709394536198078493573305, 0.02750115577417145648601708395, 1.16104461976599548219810192128, 2.19320023855886283884428783889, 2.68489429954558216776828754522, 3.37963044506288841791367393887, 4.58927072361849696084521286617, 5.498306705900766300079863050146, 6.25050401748730850069143513885, 7.00763597328495572577579550168, 7.59944426070553380593713314811, 8.569978974674355886012477238516, 9.3816756663826261273090603197, 9.71208570830299153486019536130, 10.24323832376915815878748103681, 11.10236097521460532322766173476, 11.873183507431710377802842204356, 12.62937786717823086445969942121, 13.26159666786639844612231294160, 13.67073744545809212968782076208, 14.90093882124235833153546970352, 15.74742497329422702423104572944, 16.02836562192828905241821963170, 17.01525007996222063396760442101, 17.53856630488712188080437103949, 18.01918545396909834162267162400

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