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Properties

Label 1-6017-6017.3526-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.658 - 0.753i$
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  + (−0.503 + 0.863i)2-s + (−0.858 + 0.512i)3-s + (−0.492 − 0.870i)4-s + (−0.295 + 0.955i)5-s + (−0.0103 − 0.999i)6-s + (0.998 + 0.0552i)7-s + (0.999 + 0.0138i)8-s + (0.473 − 0.880i)9-s + (−0.675 − 0.736i)10-s + (0.868 + 0.495i)12-s + (0.963 + 0.266i)13-s + (−0.550 + 0.834i)14-s + (−0.235 − 0.971i)15-s + (−0.515 + 0.856i)16-s + (−0.618 − 0.786i)17-s + (0.521 + 0.853i)18-s + ⋯
L(s)  = 1  + (−0.503 + 0.863i)2-s + (−0.858 + 0.512i)3-s + (−0.492 − 0.870i)4-s + (−0.295 + 0.955i)5-s + (−0.0103 − 0.999i)6-s + (0.998 + 0.0552i)7-s + (0.999 + 0.0138i)8-s + (0.473 − 0.880i)9-s + (−0.675 − 0.736i)10-s + (0.868 + 0.495i)12-s + (0.963 + 0.266i)13-s + (−0.550 + 0.834i)14-s + (−0.235 − 0.971i)15-s + (−0.515 + 0.856i)16-s + (−0.618 − 0.786i)17-s + (0.521 + 0.853i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.658 - 0.753i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.658 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2329061980 + 0.5128235869i\)
\(L(\frac12)\) \(\approx\) \(-0.2329061980 + 0.5128235869i\)
\(L(1)\) \(\approx\) \(0.4455471530 + 0.4197721554i\)
\(L(1)\) \(\approx\) \(0.4455471530 + 0.4197721554i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.503 + 0.863i)T \)
3 \( 1 + (-0.858 + 0.512i)T \)
5 \( 1 + (-0.295 + 0.955i)T \)
7 \( 1 + (0.998 + 0.0552i)T \)
13 \( 1 + (0.963 + 0.266i)T \)
17 \( 1 + (-0.618 - 0.786i)T \)
19 \( 1 + (-0.975 - 0.219i)T \)
23 \( 1 + (0.418 + 0.908i)T \)
29 \( 1 + (-0.498 + 0.867i)T \)
31 \( 1 + (0.882 - 0.470i)T \)
37 \( 1 + (0.367 + 0.930i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.928 + 0.370i)T \)
47 \( 1 + (-0.527 - 0.849i)T \)
53 \( 1 + (-0.380 + 0.924i)T \)
59 \( 1 + (-0.779 - 0.626i)T \)
61 \( 1 + (0.788 + 0.615i)T \)
67 \( 1 + (-0.868 - 0.495i)T \)
71 \( 1 + (0.181 + 0.983i)T \)
73 \( 1 + (-0.762 + 0.647i)T \)
79 \( 1 + (0.533 - 0.845i)T \)
83 \( 1 + (-0.681 + 0.732i)T \)
89 \( 1 + (0.999 - 0.0345i)T \)
97 \( 1 + (0.545 + 0.838i)T \)
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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

−17.35897779273341383134167692543, −16.98972390347184189003459616197, −16.240139937703633100814939300668, −15.59216438538690838679429306451, −14.60561591196965874481202821447, −13.58515658217034414820135201859, −13.10107726239834145977026725751, −12.58993864804887265902434410048, −11.93342000507249762540041092183, −11.35504192738203213473543962190, −10.74626568744805300832993879190, −10.37031076475100625791864823449, −9.19169295690753031000200270683, −8.4567104680545631730104381779, −8.18759218473501264885499211392, −7.46059105725979345229299256614, −6.452396854447269582031047508963, −5.70675223827815200376412747420, −4.733338334142689880307430631122, −4.413612668798809726125482571673, −3.6448465365072780290442684144, −2.25539538671753794969336785447, −1.741584169311604809450811690, −1.03457247577707936128560991809, −0.24308262887386059963002239846, 1.04027785219504768342637960876, 1.82388058961155239558229759532, 3.060381900191662771672935940615, 4.092666312217241956254503746985, 4.61697029225646302814150971701, 5.30285158295118401059248157069, 6.164415564031843783478725799319, 6.60674304151949710740176921614, 7.26367701607928230509283218703, 8.04345904953828835326466173168, 8.74818404250667568151363067241, 9.46959439630964823814248874096, 10.23883257545899983546713420592, 10.88368618608340690599607085771, 11.34552262125661222726067112285, 11.73185215451784312175805881966, 13.11424672190216040529079801138, 13.67875974666084085793171713849, 14.56976013559699969676583959863, 15.05069510003385143123968039535, 15.50189637029271261697094524034, 16.13523407559460150032300751973, 16.88404851946215383632969562000, 17.45835216392634159938454907038, 18.08517885367654774943423515946

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