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Properties

Label 1-4017-4017.128-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.993 + 0.116i$
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  + (0.943 − 0.332i)2-s + (0.779 − 0.626i)4-s + (0.122 − 0.992i)5-s + (−0.999 + 0.0307i)7-s + (0.526 − 0.850i)8-s + (−0.213 − 0.976i)10-s + (0.183 − 0.982i)11-s + (−0.932 + 0.361i)14-s + (0.213 − 0.976i)16-s + (0.969 + 0.243i)17-s + (−0.995 + 0.0922i)19-s + (−0.526 − 0.850i)20-s + (−0.153 − 0.988i)22-s + (0.332 − 0.943i)23-s + (−0.969 − 0.243i)25-s + ⋯
L(s)  = 1  + (0.943 − 0.332i)2-s + (0.779 − 0.626i)4-s + (0.122 − 0.992i)5-s + (−0.999 + 0.0307i)7-s + (0.526 − 0.850i)8-s + (−0.213 − 0.976i)10-s + (0.183 − 0.982i)11-s + (−0.932 + 0.361i)14-s + (0.213 − 0.976i)16-s + (0.969 + 0.243i)17-s + (−0.995 + 0.0922i)19-s + (−0.526 − 0.850i)20-s + (−0.153 − 0.988i)22-s + (0.332 − 0.943i)23-s + (−0.969 − 0.243i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.993 + 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1263409941 - 2.164200582i\)
\(L(\frac12)\) \(\approx\) \(-0.1263409941 - 2.164200582i\)
\(L(1)\) \(\approx\) \(1.273317532 - 0.9517725974i\)
\(L(1)\) \(\approx\) \(1.273317532 - 0.9517725974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.943 - 0.332i)T \)
5 \( 1 + (0.122 - 0.992i)T \)
7 \( 1 + (-0.999 + 0.0307i)T \)
11 \( 1 + (0.183 - 0.982i)T \)
17 \( 1 + (0.969 + 0.243i)T \)
19 \( 1 + (-0.995 + 0.0922i)T \)
23 \( 1 + (0.332 - 0.943i)T \)
29 \( 1 + (-0.992 - 0.122i)T \)
31 \( 1 + (-0.673 + 0.739i)T \)
37 \( 1 + (0.833 - 0.552i)T \)
41 \( 1 + (0.920 + 0.389i)T \)
43 \( 1 + (-0.445 - 0.895i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.816 + 0.577i)T \)
59 \( 1 + (0.526 - 0.850i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (0.526 - 0.850i)T \)
71 \( 1 + (-0.122 - 0.992i)T \)
73 \( 1 + (-0.798 + 0.602i)T \)
79 \( 1 + (-0.602 + 0.798i)T \)
83 \( 1 + (-0.999 + 0.0307i)T \)
89 \( 1 + (-0.988 - 0.153i)T \)
97 \( 1 + (-0.961 - 0.273i)T \)
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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

−18.95462062856453725306715085337, −18.110611616649762761392295796, −17.23378776785716558628097959253, −16.77715667897662880271963016, −15.89985952618751745595826613001, −15.2873983490366573541522513137, −14.69467295687325313205156304763, −14.26380291844499295267653178426, −13.165129927806091373532591214079, −12.977183796743692432361164238346, −12.0933714386811412160805993054, −11.36796353751454525913619953787, −10.705797539558275131168828836520, −9.83176839739063898475374565700, −9.35373674558878649843018315826, −8.04659416436762933882427646450, −7.22644495724769600773472776871, −7.01396288774419577791554153359, −5.98003251994231578434369778845, −5.705069907656365927408032644949, −4.474321136566946618560093055388, −3.83026342638157149605229456583, −3.09508447471958118047925422705, −2.475926094785855460617435935387, −1.58042154826269996249188933275, 0.39817619625286734070949613158, 1.28508826067109661049880191154, 2.25209130854598877413607500202, 3.132437206384334737336097330098, 3.83771167480785777463352964389, 4.445797275719110079271927036863, 5.53715424739608508109800160139, 5.83651440664753227476512145959, 6.593365142646161209657175866459, 7.499332899219456627284984950169, 8.48695076774418627620154726926, 9.16650765721809943157991828523, 9.90015714815249557057400730364, 10.669188934574236821388821165435, 11.31008774419741761124839291567, 12.373506830354704846649084669460, 12.57820131436470271469731110633, 13.193596713544506517329780092756, 13.919980981514592217188227505141, 14.57630370968583918621326694561, 15.384001576954009223897168120304, 16.14054729640060688244172099416, 16.68405238421945671440232188830, 16.958563411052475178715561972879, 18.433526049855565219823776286055

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