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Properties

Label 1-4008-4008.485-r0-0-0
Degree $1$
Conductor $4008$
Sign $-0.418 - 0.908i$
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  + (−0.974 + 0.225i)5-s + (−0.0944 − 0.995i)7-s + (0.997 − 0.0756i)11-s + (−0.169 + 0.985i)13-s + (0.862 + 0.505i)17-s + (−0.822 + 0.569i)19-s + (−0.881 + 0.472i)23-s + (0.898 − 0.438i)25-s + (−0.881 − 0.472i)29-s + (−0.0189 − 0.999i)31-s + (0.316 + 0.948i)35-s + (0.280 − 0.959i)37-s + (−0.999 + 0.0378i)41-s + (0.614 − 0.788i)43-s + (−0.132 − 0.991i)47-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)5-s + (−0.0944 − 0.995i)7-s + (0.997 − 0.0756i)11-s + (−0.169 + 0.985i)13-s + (0.862 + 0.505i)17-s + (−0.822 + 0.569i)19-s + (−0.881 + 0.472i)23-s + (0.898 − 0.438i)25-s + (−0.881 − 0.472i)29-s + (−0.0189 − 0.999i)31-s + (0.316 + 0.948i)35-s + (0.280 − 0.959i)37-s + (−0.999 + 0.0378i)41-s + (0.614 − 0.788i)43-s + (−0.132 − 0.991i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.418 - 0.908i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ -0.418 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3762677552 - 0.5876406466i\)
\(L(\frac12)\) \(\approx\) \(0.3762677552 - 0.5876406466i\)
\(L(1)\) \(\approx\) \(0.8118414362 - 0.07601807604i\)
\(L(1)\) \(\approx\) \(0.8118414362 - 0.07601807604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.974 + 0.225i)T \)
7 \( 1 + (-0.0944 - 0.995i)T \)
11 \( 1 + (0.997 - 0.0756i)T \)
13 \( 1 + (-0.169 + 0.985i)T \)
17 \( 1 + (0.862 + 0.505i)T \)
19 \( 1 + (-0.822 + 0.569i)T \)
23 \( 1 + (-0.881 + 0.472i)T \)
29 \( 1 + (-0.881 - 0.472i)T \)
31 \( 1 + (-0.0189 - 0.999i)T \)
37 \( 1 + (0.280 - 0.959i)T \)
41 \( 1 + (-0.999 + 0.0378i)T \)
43 \( 1 + (0.614 - 0.788i)T \)
47 \( 1 + (-0.132 - 0.991i)T \)
53 \( 1 + (-0.929 - 0.369i)T \)
59 \( 1 + (-0.862 + 0.505i)T \)
61 \( 1 + (0.752 + 0.658i)T \)
67 \( 1 + (0.974 + 0.225i)T \)
71 \( 1 + (-0.700 + 0.713i)T \)
73 \( 1 + (-0.206 + 0.978i)T \)
79 \( 1 + (0.584 + 0.811i)T \)
83 \( 1 + (0.942 - 0.334i)T \)
89 \( 1 + (0.914 - 0.404i)T \)
97 \( 1 + (-0.0189 + 0.999i)T \)
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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

−18.90532435782460365884789880987, −18.041961976935429706294287048891, −17.35874954556236766594813049983, −16.51332175410411667189755011003, −15.994073178760572514107480102779, −15.2281454827767025634422971407, −14.79428729662701301998480803854, −14.11070231427259562507433621706, −12.959398292336251689675808876181, −12.409450678259520821865618663317, −11.977841483671333297005422920, −11.25239860661223569474911907157, −10.51001717944270200925255027079, −9.51321431408298953746984196947, −8.990864943046907953875184243238, −8.1674495251716903163998006256, −7.7258801234931590739168029705, −6.6842616274884798319856406548, −6.093068097515746457245787242917, −5.07866086556770829816493808936, −4.60051316122703368403131093086, −3.47276428910526813259137378954, −3.0719156887434047583829484015, −1.96695281837498630429224106458, −0.95857976805783883638089699077, 0.23277073717370691514850048391, 1.37091931458152602207492819805, 2.17735715290060189025858122954, 3.578621496593565315742126860667, 3.90705383861749330842638984364, 4.34448390774088782968930769987, 5.61444680948910773136315736980, 6.44459731522677871374624533813, 7.07919173097386569321290285953, 7.7407396831370835232112350493, 8.35927343127041670857859931499, 9.30150239230829419203710789177, 9.998508003429255461911507622561, 10.72663526270768736760372673883, 11.50512916873740437679435209093, 11.92421633051556634970414839802, 12.713580303815767239079888692724, 13.561407881167781137815406161469, 14.41687370469031992483969733811, 14.6283557887240695072556759931, 15.55762114774277875563071123736, 16.42150758217770653144143461093, 16.8301501337342268429880966261, 17.32541440747545958939585774711, 18.476699972193591526126775539971

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