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Properties

Label 1-4008-4008.635-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.312 - 0.949i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (0.0944 + 0.995i)5-s + (0.169 + 0.985i)7-s + (0.881 + 0.472i)11-s + (−0.316 − 0.948i)13-s + (0.954 − 0.298i)17-s + (−0.700 − 0.713i)19-s + (−0.0567 + 0.998i)23-s + (−0.982 + 0.188i)25-s + (0.0567 + 0.998i)29-s + (−0.614 − 0.788i)31-s + (−0.965 + 0.261i)35-s + (0.489 − 0.872i)37-s + (−0.243 + 0.969i)41-s + (−0.929 + 0.369i)43-s + (0.997 + 0.0756i)47-s + ⋯
L(s)  = 1  + (0.0944 + 0.995i)5-s + (0.169 + 0.985i)7-s + (0.881 + 0.472i)11-s + (−0.316 − 0.948i)13-s + (0.954 − 0.298i)17-s + (−0.700 − 0.713i)19-s + (−0.0567 + 0.998i)23-s + (−0.982 + 0.188i)25-s + (0.0567 + 0.998i)29-s + (−0.614 − 0.788i)31-s + (−0.965 + 0.261i)35-s + (0.489 − 0.872i)37-s + (−0.243 + 0.969i)41-s + (−0.929 + 0.369i)43-s + (0.997 + 0.0756i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.312 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8091696055 - 0.5854274365i\)
\(L(\frac12)\) \(\approx\) \(0.8091696055 - 0.5854274365i\)
\(L(1)\) \(\approx\) \(1.012715470 + 0.2328263757i\)
\(L(1)\) \(\approx\) \(1.012715470 + 0.2328263757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.0944 + 0.995i)T \)
7 \( 1 + (0.169 + 0.985i)T \)
11 \( 1 + (0.881 + 0.472i)T \)
13 \( 1 + (-0.316 - 0.948i)T \)
17 \( 1 + (0.954 - 0.298i)T \)
19 \( 1 + (-0.700 - 0.713i)T \)
23 \( 1 + (-0.0567 + 0.998i)T \)
29 \( 1 + (0.0567 + 0.998i)T \)
31 \( 1 + (-0.614 - 0.788i)T \)
37 \( 1 + (0.489 - 0.872i)T \)
41 \( 1 + (-0.243 + 0.969i)T \)
43 \( 1 + (-0.929 + 0.369i)T \)
47 \( 1 + (0.997 + 0.0756i)T \)
53 \( 1 + (-0.776 - 0.629i)T \)
59 \( 1 + (0.954 + 0.298i)T \)
61 \( 1 + (0.999 + 0.0378i)T \)
67 \( 1 + (0.0944 - 0.995i)T \)
71 \( 1 + (-0.898 - 0.438i)T \)
73 \( 1 + (0.843 - 0.537i)T \)
79 \( 1 + (0.132 - 0.991i)T \)
83 \( 1 + (-0.800 - 0.599i)T \)
89 \( 1 + (-0.421 - 0.906i)T \)
97 \( 1 + (0.614 - 0.788i)T \)
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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

−18.65309204284554286964680306773, −17.3719819769053367883304499006, −17.02003836487556923103224340273, −16.59346786167764369228240663313, −16.02078664709057471457818166424, −14.86065411684612957064664342857, −14.243165623433149086471307797055, −13.79905240618623440511782266220, −12.94754868045341243866607249173, −12.23287477865453136467447087950, −11.736783023601204423684391905731, −10.846699700057473582079589975982, −10.04424378751204544493747209303, −9.51719606395181447951589995484, −8.50447991771033750277122132608, −8.2514968939013481393387045938, −7.17165880459919944522320923030, −6.52746251050401357957272688791, −5.710996758471825550200373628, −4.86413911660200578438406257328, −3.98837093598652823997310845515, −3.82102921514043750505307986409, −2.35474558758289510145366882405, −1.424832905207295933049003267206, −0.90778739649369371788317489271, 0.15570927189237790784982938532, 1.464031213319627386112394535874, 2.246892063584250079469040456684, 3.03052701099703933325856286242, 3.63144490486785014125220307543, 4.741991195089113486412811308859, 5.54858298651910257167407731907, 6.12078103369705634161140266265, 6.99095326139028850512701683347, 7.58328990845112986675607971561, 8.37913317896651019863276388095, 9.352700430322909161692561345355, 9.73630475908756321715280272959, 10.632270558247894409323563945672, 11.363237569684169318118642928643, 11.890765439775174447169061462184, 12.67636905686388789209243725148, 13.33416230964478548039276454421, 14.43657284117968665195327147280, 14.73996430166598551306595720381, 15.24495327444224226818650008893, 16.019060933784136228815454668773, 16.94812724728884201723745022994, 17.68552896548305991902026972010, 18.10344514931353348654650956241

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