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Properties

Label 1-4008-4008.971-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.951 - 0.307i$
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  + (0.965 + 0.261i)5-s + (−0.988 + 0.150i)7-s + (−0.421 − 0.906i)11-s + (0.553 − 0.832i)13-s + (−0.0944 − 0.995i)17-s + (−0.942 + 0.334i)19-s + (0.455 + 0.890i)23-s + (0.862 + 0.505i)25-s + (−0.455 + 0.890i)29-s + (−0.280 + 0.959i)31-s + (−0.993 − 0.113i)35-s + (0.898 + 0.438i)37-s + (−0.843 − 0.537i)41-s + (−0.489 − 0.872i)43-s + (−0.914 − 0.404i)47-s + ⋯
L(s)  = 1  + (0.965 + 0.261i)5-s + (−0.988 + 0.150i)7-s + (−0.421 − 0.906i)11-s + (0.553 − 0.832i)13-s + (−0.0944 − 0.995i)17-s + (−0.942 + 0.334i)19-s + (0.455 + 0.890i)23-s + (0.862 + 0.505i)25-s + (−0.455 + 0.890i)29-s + (−0.280 + 0.959i)31-s + (−0.993 − 0.113i)35-s + (0.898 + 0.438i)37-s + (−0.843 − 0.537i)41-s + (−0.489 − 0.872i)43-s + (−0.914 − 0.404i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.905284368 - 0.3004255704i\)
\(L(\frac12)\) \(\approx\) \(1.905284368 - 0.3004255704i\)
\(L(1)\) \(\approx\) \(1.059500530 + 0.02697335861i\)
\(L(1)\) \(\approx\) \(1.059500530 + 0.02697335861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.965 + 0.261i)T \)
7 \( 1 + (-0.988 + 0.150i)T \)
11 \( 1 + (-0.421 - 0.906i)T \)
13 \( 1 + (0.553 - 0.832i)T \)
17 \( 1 + (-0.0944 - 0.995i)T \)
19 \( 1 + (-0.942 + 0.334i)T \)
23 \( 1 + (0.455 + 0.890i)T \)
29 \( 1 + (-0.455 + 0.890i)T \)
31 \( 1 + (-0.280 + 0.959i)T \)
37 \( 1 + (0.898 + 0.438i)T \)
41 \( 1 + (-0.843 - 0.537i)T \)
43 \( 1 + (-0.489 - 0.872i)T \)
47 \( 1 + (-0.914 - 0.404i)T \)
53 \( 1 + (-0.822 - 0.569i)T \)
59 \( 1 + (-0.0944 + 0.995i)T \)
61 \( 1 + (-0.206 + 0.978i)T \)
67 \( 1 + (0.965 - 0.261i)T \)
71 \( 1 + (0.800 - 0.599i)T \)
73 \( 1 + (0.0189 - 0.999i)T \)
79 \( 1 + (0.0567 + 0.998i)T \)
83 \( 1 + (-0.387 + 0.922i)T \)
89 \( 1 + (0.999 - 0.0378i)T \)
97 \( 1 + (0.280 + 0.959i)T \)
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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

−18.54831158091152740548726014309, −17.45986049067032755981838312775, −17.0270213491993364031240134606, −16.46611385032791206058942436105, −15.64853648579891936960434109930, −14.9056348690322473027660751673, −14.28828814585996660070554133689, −13.2632170461087931799542056780, −12.975475891017176150610505735122, −12.53586027693781920158012937022, −11.33003093641060327542869210484, −10.705852221405041075116497011972, −9.78004454181026671005292124088, −9.59776663336241696467752153781, −8.68796925227662693091306686303, −7.96242737119230984228277843602, −6.825220532502599282622128602278, −6.38055286634685875667500308612, −5.84655670644938845339264105, −4.681262665657012490202647839080, −4.229740219148519384580216666218, −3.16281893770097681513978235885, −2.2163865607002369479830324285, −1.740466877186177877160981723820, −0.52300796346586626111276549022, 0.45395623018677502598862908105, 1.42340134220031647562809936168, 2.44642996696541912382134613040, 3.18817160478367765864855516316, 3.61022025995932900055961209503, 5.1728096090436804314494730034, 5.443126759858528675342979921279, 6.37660540706351364863977924016, 6.7869827594568321345959223750, 7.800432266364103594382333063683, 8.736989841623941745795715954410, 9.20680209855876974397441451965, 10.06498659143433659844782837323, 10.58631546419007171028493141695, 11.22454835072390033860465936172, 12.23546364043247074840768928307, 13.09322139761836377413854827673, 13.34699366904943940287040487261, 14.022658513090157024133207716290, 14.93345670852664706787653839633, 15.57572732341033450502949133272, 16.355480951305967642479961811991, 16.79206885365502127713928901047, 17.68623044303835960124840266103, 18.411139482762482270447338159975

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