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Properties

Label 1-4017-4017.1598-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.551 - 0.833i$
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  + (0.552 − 0.833i)2-s + (−0.389 − 0.920i)4-s + (0.952 − 0.303i)5-s + (−0.332 + 0.943i)7-s + (−0.982 − 0.183i)8-s + (0.273 − 0.961i)10-s + (0.998 + 0.0615i)11-s + (0.602 + 0.798i)14-s + (−0.696 + 0.717i)16-s + (0.908 + 0.417i)17-s + (−0.881 − 0.473i)19-s + (−0.650 − 0.759i)20-s + (0.602 − 0.798i)22-s + (−0.445 + 0.895i)23-s + (0.816 − 0.577i)25-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + (−0.389 − 0.920i)4-s + (0.952 − 0.303i)5-s + (−0.332 + 0.943i)7-s + (−0.982 − 0.183i)8-s + (0.273 − 0.961i)10-s + (0.998 + 0.0615i)11-s + (0.602 + 0.798i)14-s + (−0.696 + 0.717i)16-s + (0.908 + 0.417i)17-s + (−0.881 − 0.473i)19-s + (−0.650 − 0.759i)20-s + (0.602 − 0.798i)22-s + (−0.445 + 0.895i)23-s + (0.816 − 0.577i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.551 - 0.833i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (1598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.551 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.425254378 - 1.303416630i\)
\(L(\frac12)\) \(\approx\) \(2.425254378 - 1.303416630i\)
\(L(1)\) \(\approx\) \(1.458176149 - 0.6569870629i\)
\(L(1)\) \(\approx\) \(1.458176149 - 0.6569870629i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.552 - 0.833i)T \)
5 \( 1 + (0.952 - 0.303i)T \)
7 \( 1 + (-0.332 + 0.943i)T \)
11 \( 1 + (0.998 + 0.0615i)T \)
17 \( 1 + (0.908 + 0.417i)T \)
19 \( 1 + (-0.881 - 0.473i)T \)
23 \( 1 + (-0.445 + 0.895i)T \)
29 \( 1 + (-0.213 + 0.976i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 + (0.932 - 0.361i)T \)
41 \( 1 + (-0.952 - 0.303i)T \)
43 \( 1 + (0.153 + 0.988i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.0307 - 0.999i)T \)
59 \( 1 + (0.332 + 0.943i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (0.650 - 0.759i)T \)
71 \( 1 + (-0.213 - 0.976i)T \)
73 \( 1 + (0.739 - 0.673i)T \)
79 \( 1 + (0.739 + 0.673i)T \)
83 \( 1 + (0.650 + 0.759i)T \)
89 \( 1 + (0.602 + 0.798i)T \)
97 \( 1 + (0.908 - 0.417i)T \)
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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

−18.60266686252074193901919913064, −17.454659971756481126492190140294, −17.18866745115840702518154358519, −16.61853625390441203351335409930, −16.00407706409773267082537218596, −14.96331932657015959587417999158, −14.28164002182648638069288365694, −14.0942555348963739496267091313, −13.292617592894442553096971742326, −12.63644386680388016225047861366, −11.95621423827625537531651619369, −10.98643748675877024802783753371, −10.137180981672807941753899307425, −9.592594288828896069181392049145, −8.78436796945981140154283392183, −7.95946464939489423676131827303, −7.17023731168423069891908824617, −6.45398328890281883666808314259, −6.14865788636410147680663017522, −5.21037522660030879500078558303, −4.345587724558904881482452961356, −3.70727809226582394827767819589, −2.931006967583765149016043421812, −1.92552558384592889056003294257, −0.77985736914970382928791175567, 0.88786637624309202426932099417, 1.827634159759292248525062524883, 2.25916834760352867807380509112, 3.27564408573604332895534079309, 3.94153050798146110406800116334, 4.934259917201405965448297106579, 5.59349049207330261005472613092, 6.1343779553212574490427129226, 6.77133950876610605566131795142, 8.15024113520256500571601019924, 9.08499917474003610203454146723, 9.341503784186586415051171250870, 10.060007509046544359565193349480, 10.81939038864887392997615173646, 11.715375017104844170617599732289, 12.1870430929349710050133150782, 12.94043284415931380085794519178, 13.35673751990403827679713853169, 14.26721948925735973505757241117, 14.7955568494905661368509211914, 15.34259848015702343318671816467, 16.47776203681395721009103613285, 16.98649162237755539748544882202, 18.00130163437713097825502748472, 18.30656346454276633789386150270

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