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Properties

Label 1-4008-4008.1133-r0-0-0
Degree $1$
Conductor $4008$
Sign $-0.999 - 0.0395i$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
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  + (0.752 + 0.658i)5-s + (−0.999 − 0.0378i)7-s + (0.280 + 0.959i)11-s + (−0.243 + 0.969i)13-s + (−0.914 − 0.404i)17-s + (0.644 + 0.764i)19-s + (0.489 − 0.872i)23-s + (0.132 + 0.991i)25-s + (0.489 + 0.872i)29-s + (−0.316 + 0.948i)31-s + (−0.726 − 0.686i)35-s + (−0.993 + 0.113i)37-s + (−0.800 − 0.599i)41-s + (−0.965 + 0.261i)43-s + (−0.776 − 0.629i)47-s + ⋯
L(s)  = 1  + (0.752 + 0.658i)5-s + (−0.999 − 0.0378i)7-s + (0.280 + 0.959i)11-s + (−0.243 + 0.969i)13-s + (−0.914 − 0.404i)17-s + (0.644 + 0.764i)19-s + (0.489 − 0.872i)23-s + (0.132 + 0.991i)25-s + (0.489 + 0.872i)29-s + (−0.316 + 0.948i)31-s + (−0.726 − 0.686i)35-s + (−0.993 + 0.113i)37-s + (−0.800 − 0.599i)41-s + (−0.965 + 0.261i)43-s + (−0.776 − 0.629i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.999 - 0.0395i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ -0.999 - 0.0395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01577498148 + 0.7982990788i\)
\(L(\frac12)\) \(\approx\) \(0.01577498148 + 0.7982990788i\)
\(L(1)\) \(\approx\) \(0.8782987116 + 0.3192703355i\)
\(L(1)\) \(\approx\) \(0.8782987116 + 0.3192703355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.752 + 0.658i)T \)
7 \( 1 + (-0.999 - 0.0378i)T \)
11 \( 1 + (0.280 + 0.959i)T \)
13 \( 1 + (-0.243 + 0.969i)T \)
17 \( 1 + (-0.914 - 0.404i)T \)
19 \( 1 + (0.644 + 0.764i)T \)
23 \( 1 + (0.489 - 0.872i)T \)
29 \( 1 + (0.489 + 0.872i)T \)
31 \( 1 + (-0.316 + 0.948i)T \)
37 \( 1 + (-0.993 + 0.113i)T \)
41 \( 1 + (-0.800 - 0.599i)T \)
43 \( 1 + (-0.965 + 0.261i)T \)
47 \( 1 + (-0.776 - 0.629i)T \)
53 \( 1 + (-0.988 + 0.150i)T \)
59 \( 1 + (0.914 - 0.404i)T \)
61 \( 1 + (0.942 + 0.334i)T \)
67 \( 1 + (-0.752 + 0.658i)T \)
71 \( 1 + (-0.584 - 0.811i)T \)
73 \( 1 + (0.387 + 0.922i)T \)
79 \( 1 + (-0.929 + 0.369i)T \)
83 \( 1 + (0.881 - 0.472i)T \)
89 \( 1 + (0.700 + 0.713i)T \)
97 \( 1 + (-0.316 - 0.948i)T \)
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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

−17.979539079981371095857834951346, −17.368622090371041727494956055594, −16.90734566014188416308582580056, −16.01285050679703823829916152774, −15.628949272638644457551002933920, −14.78651804794940256902628966241, −13.73886201158202678043745014248, −13.23482760986535614726380886026, −13.0151391376004083366985824607, −11.9629616688550138615048499754, −11.31147883161414326722495288985, −10.37781910937575262702321953386, −9.776156967107408642194831066930, −9.1321636322027802938790358249, −8.5536655045939025242202533006, −7.703938453824606127765544029019, −6.66440971585613447999811584415, −6.15543629599436675092669356360, −5.42303813393210332637997086301, −4.77337950518142101450709179650, −3.63732636178281624933796505390, −3.03498602624274105363921948191, −2.1687888131673820563427289208, −1.10770362352806206450983586628, −0.22246220825275781751293987029, 1.46809851839297313535446022150, 2.116562461492626757273652979204, 2.995972719822573851781804034069, 3.658381030737948009337973004044, 4.70292322684146682685974103027, 5.35031861633371779021100901619, 6.5064338720650951581054186696, 6.780910830375902936473698193848, 7.25058662061024570566591144671, 8.66898917223158922204906071293, 9.15378985239089020502014670102, 10.05374814585292603573948931821, 10.203607857222651040543954522728, 11.255679922686874293046591326938, 12.04406338909437899447904319832, 12.68619822664149094999650349357, 13.407321406314908956547132955651, 14.12929717801079778461562329784, 14.59516595198115118749298429740, 15.411055671225504973251217577166, 16.22190193485827099817384144633, 16.71625407855573314657186619863, 17.595174533673030329191440109748, 18.08828116034491767837942418976, 18.83632585937037317749891849515

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