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Properties

Label 1-4017-4017.2690-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.982 + 0.188i$
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  + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (0.0307 + 0.999i)5-s + (−0.992 + 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.982 + 0.183i)10-s + (0.952 − 0.303i)11-s + (−0.0922 + 0.995i)14-s + (0.650 + 0.759i)16-s + (−0.552 − 0.833i)17-s + (0.779 + 0.626i)19-s + (0.389 − 0.920i)20-s + (−0.0922 − 0.995i)22-s + (−0.739 + 0.673i)23-s + (−0.998 + 0.0615i)25-s + ⋯
L(s)  = 1  + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (0.0307 + 0.999i)5-s + (−0.992 + 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.982 + 0.183i)10-s + (0.952 − 0.303i)11-s + (−0.0922 + 0.995i)14-s + (0.650 + 0.759i)16-s + (−0.552 − 0.833i)17-s + (0.779 + 0.626i)19-s + (0.389 − 0.920i)20-s + (−0.0922 − 0.995i)22-s + (−0.739 + 0.673i)23-s + (−0.998 + 0.0615i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.982 + 0.188i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2690, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.982 + 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02263180849 - 0.2384896072i\)
\(L(\frac12)\) \(\approx\) \(0.02263180849 - 0.2384896072i\)
\(L(1)\) \(\approx\) \(0.7727211008 - 0.2640623045i\)
\(L(1)\) \(\approx\) \(0.7727211008 - 0.2640623045i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.213 - 0.976i)T \)
5 \( 1 + (0.0307 + 0.999i)T \)
7 \( 1 + (-0.992 + 0.122i)T \)
11 \( 1 + (0.952 - 0.303i)T \)
17 \( 1 + (-0.552 - 0.833i)T \)
19 \( 1 + (0.779 + 0.626i)T \)
23 \( 1 + (-0.739 + 0.673i)T \)
29 \( 1 + (-0.881 - 0.473i)T \)
31 \( 1 + (-0.982 + 0.183i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (-0.0307 + 0.999i)T \)
43 \( 1 + (0.696 - 0.717i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.153 + 0.988i)T \)
59 \( 1 + (0.992 + 0.122i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (-0.389 - 0.920i)T \)
71 \( 1 + (-0.881 + 0.473i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (-0.389 + 0.920i)T \)
89 \( 1 + (-0.0922 + 0.995i)T \)
97 \( 1 + (-0.552 + 0.833i)T \)
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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

−18.82801303292923012481063322382, −17.71220766276069039754663207449, −17.46586944269107073146171210259, −16.59644768323437871426569664941, −16.14437183376386155467536570441, −15.70041211893287785073680473128, −14.705597205128444869312632289115, −14.2131338771494041891651702335, −13.22949431966685219404610412072, −12.90033407075491959065725396668, −12.30791397949932672960563929411, −11.49429022181767187054037952077, −10.301106188924384794206852903435, −9.52045011240679402743103624531, −8.99260012306427110173877493974, −8.56699949804933077114844603137, −7.388217175596303519354735884400, −7.04169502520423248740074531554, −5.94002366483674938699145229901, −5.73690035797683445959348452377, −4.56808861996712435292979819002, −4.05476159463149676367082815297, −3.40930501152089494338419205370, −2.107507122313707021874525378529, −0.974490827428425641490550472985, 0.07244452914475693936356008388, 1.36807040032404172745489483839, 2.24815040664190074067780793242, 3.050546736766634595165484084249, 3.60805772077189194798372485003, 4.17261851848471464101414062956, 5.470101876489589401630747321574, 5.97235895620397482672483026931, 6.80177378870600293042464203532, 7.52889047019953162040877741505, 8.63032941480105402270275943823, 9.51484830136153250152920238936, 9.72028469881535246231673466376, 10.57137932339298713805478267132, 11.37810596251308540611074370394, 11.78916174756764768227934571704, 12.49159214436540054734893620255, 13.54952802668803133043052070845, 13.74257133465791557266085477608, 14.55091469643291119254820203316, 15.24045073775742087306586832103, 16.0133687074462059804917698819, 16.85954744247890897441857078791, 17.67678711498998225044355556368, 18.44443694514387454306963465796

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