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Properties

Label 1-4017-4017.3587-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.812 + 0.582i$
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  + (−0.696 − 0.717i)2-s + (−0.0307 + 0.999i)4-s + (−0.332 − 0.943i)5-s + (−0.213 + 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (−0.969 + 0.243i)11-s + (0.850 − 0.526i)14-s + (−0.998 − 0.0615i)16-s + (0.153 + 0.988i)17-s + (0.389 + 0.920i)19-s + (0.952 − 0.303i)20-s + (0.850 + 0.526i)22-s + (0.273 + 0.961i)23-s + (−0.779 + 0.626i)25-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)2-s + (−0.0307 + 0.999i)4-s + (−0.332 − 0.943i)5-s + (−0.213 + 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (−0.969 + 0.243i)11-s + (0.850 − 0.526i)14-s + (−0.998 − 0.0615i)16-s + (0.153 + 0.988i)17-s + (0.389 + 0.920i)19-s + (0.952 − 0.303i)20-s + (0.850 + 0.526i)22-s + (0.273 + 0.961i)23-s + (−0.779 + 0.626i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.812 + 0.582i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (3587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.812 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1016020254 + 0.3159795406i\)
\(L(\frac12)\) \(\approx\) \(0.1016020254 + 0.3159795406i\)
\(L(1)\) \(\approx\) \(0.5804871359 - 0.04269915995i\)
\(L(1)\) \(\approx\) \(0.5804871359 - 0.04269915995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.696 - 0.717i)T \)
5 \( 1 + (-0.332 - 0.943i)T \)
7 \( 1 + (-0.213 + 0.976i)T \)
11 \( 1 + (-0.969 + 0.243i)T \)
17 \( 1 + (0.153 + 0.988i)T \)
19 \( 1 + (0.389 + 0.920i)T \)
23 \( 1 + (0.273 + 0.961i)T \)
29 \( 1 + (-0.650 + 0.759i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (0.0922 + 0.995i)T \)
41 \( 1 + (0.332 - 0.943i)T \)
43 \( 1 + (-0.816 - 0.577i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.992 + 0.122i)T \)
59 \( 1 + (0.213 + 0.976i)T \)
61 \( 1 + (0.932 + 0.361i)T \)
67 \( 1 + (-0.952 - 0.303i)T \)
71 \( 1 + (-0.650 - 0.759i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.982 - 0.183i)T \)
83 \( 1 + (-0.952 + 0.303i)T \)
89 \( 1 + (0.850 - 0.526i)T \)
97 \( 1 + (0.153 - 0.988i)T \)
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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

−18.14349849575042346092878600671, −17.64074844000767402125878689120, −16.77173871052215629959702910780, −16.152349532995177402352001315667, −15.65361186491121377012074166093, −14.85220989732249627728510188512, −14.33133786444005067236922021050, −13.47477783040054647887948558583, −13.14264498766069915087978777457, −11.56972108916997000314676699381, −11.24327867626812631366757868969, −10.38613401601751469008199406500, −10.00983206303660791176886340383, −9.21140471263768069875081596438, −8.21041291489670851215873921797, −7.59115855606379569128410518376, −7.12091991969610254147543888680, −6.49662951809316309006874926362, −5.64443685713996375683631931963, −4.782646376655463653139938667960, −4.00207145926925945490787555624, −2.877928819356105855630568035998, −2.3169063577914421600615866931, −0.84616601953126413043072713963, −0.15366029963255324506098775816, 1.2983863537630188828315830865, 1.8063971082180439675084602832, 2.86141312740238228000242984448, 3.520100712850154488648804637849, 4.40387266722694634612634180261, 5.31437852139973502725603228725, 5.85952384819152128760647326129, 7.19736923567704347172369852615, 7.77058630705437429505670471208, 8.68899098467756560667626737592, 8.76833518119026061917736083095, 9.86616680017934362066493840715, 10.28429855236247973248619765368, 11.25073658039219506087844849242, 12.01462332849724589273364984271, 12.39314874171298423524444892865, 13.02017686940199347749590179886, 13.61911357860424304989856223432, 14.88911856107783945350848351498, 15.57514633698913616526755898304, 16.131434680007835237784083586036, 16.76191745918661827747751531223, 17.51534308991737732548080913839, 18.138681440151106886441361490145, 18.88009035057406168890098467551

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