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Properties

Label 1-6017-6017.1700-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.794 - 0.606i$
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  + (0.513 − 0.857i)2-s + (−0.309 − 0.951i)3-s + (−0.471 − 0.881i)4-s + (0.704 + 0.709i)5-s + (−0.974 − 0.223i)6-s + (−0.324 − 0.945i)7-s + (−0.998 − 0.0483i)8-s + (−0.809 + 0.587i)9-s + (0.970 − 0.239i)10-s + (−0.692 + 0.721i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.457 − 0.889i)15-s + (−0.554 + 0.832i)16-s + (−0.877 + 0.478i)17-s + (0.0884 + 0.996i)18-s + ⋯
L(s)  = 1  + (0.513 − 0.857i)2-s + (−0.309 − 0.951i)3-s + (−0.471 − 0.881i)4-s + (0.704 + 0.709i)5-s + (−0.974 − 0.223i)6-s + (−0.324 − 0.945i)7-s + (−0.998 − 0.0483i)8-s + (−0.809 + 0.587i)9-s + (0.970 − 0.239i)10-s + (−0.692 + 0.721i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.457 − 0.889i)15-s + (−0.554 + 0.832i)16-s + (−0.877 + 0.478i)17-s + (0.0884 + 0.996i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.794 - 0.606i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1700, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5888234803 - 1.741471512i\)
\(L(\frac12)\) \(\approx\) \(0.5888234803 - 1.741471512i\)
\(L(1)\) \(\approx\) \(0.8319035992 - 0.8418456475i\)
\(L(1)\) \(\approx\) \(0.8319035992 - 0.8418456475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.513 - 0.857i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.704 + 0.709i)T \)
7 \( 1 + (-0.324 - 0.945i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.877 + 0.478i)T \)
19 \( 1 + (0.962 - 0.270i)T \)
23 \( 1 + (0.200 + 0.979i)T \)
29 \( 1 + (0.861 - 0.506i)T \)
31 \( 1 + (0.989 - 0.144i)T \)
37 \( 1 + (-0.870 - 0.493i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (0.594 + 0.804i)T \)
53 \( 1 + (0.0563 - 0.998i)T \)
59 \( 1 + (0.962 + 0.270i)T \)
61 \( 1 + (0.974 + 0.223i)T \)
67 \( 1 + (0.278 + 0.960i)T \)
71 \( 1 + (-0.369 - 0.929i)T \)
73 \( 1 + (-0.152 + 0.988i)T \)
79 \( 1 + (0.926 - 0.377i)T \)
83 \( 1 + (0.966 - 0.254i)T \)
89 \( 1 + (-0.120 - 0.992i)T \)
97 \( 1 + (0.759 - 0.650i)T \)
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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

−17.70435783380106035220040538587, −17.16283307158950337521618899866, −16.43784748556431827426786399614, −16.0577302807796066118146636272, −15.47450957151007089594223703937, −14.81919951807744390608913584803, −14.08754219759203548116982042885, −13.63112717711144195851927701408, −12.64656358286124227200762693822, −12.082809467946750789862912983283, −11.759393064435538550705404044004, −10.527664482682891222781528739443, −9.750639216823985404646866419590, −9.2223353824755368921964113275, −8.754373341316066418286510945969, −8.10521075061236140188414732244, −6.82608888623853231987680216001, −6.43479673108476558484614397471, −5.58094395801046515564763648951, −5.02179074559469412848080036469, −4.71722596375849713592398731433, −3.80558910140424050226871187909, −2.77688897576306321287949620482, −2.35702002265170399468739101279, −0.703446802291398133083043921034, 0.58603110900992692645255774758, 1.32350073996466117742218124217, 2.228541774271764582221593192628, 2.74361464310133456021705177138, 3.471431631052313574556355256, 4.44405956696324903580553284560, 5.27363436221898189184131710233, 5.88327960415751401712711199939, 6.64073329428439510584432641135, 7.13265636281945102773581303489, 7.889624729701789151986347730687, 8.98420234647262543087340322102, 9.78700178749204575533728499910, 10.274476929583207888855255895472, 10.94123444662500996289385639622, 11.5176480812123643363028126076, 12.19784962511059961820253025216, 13.12192228811702343915457438947, 13.2612299307785034275399652859, 14.07974096159288898144893935128, 14.34542099796820535262621477454, 15.31894804161614413177258499072, 16.094119908388344065534132036272, 17.40630109336938008517871477211, 17.52932128626047031246398061117

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