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Properties

Label 1-4007-4007.60-r1-0-0
Degree $1$
Conductor $4007$
Sign $-0.690 - 0.723i$
Analytic cond. $430.611$
Root an. cond. $430.611$
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  + (−0.755 − 0.654i)2-s + (0.337 + 0.941i)3-s + (0.143 + 0.989i)4-s + (−0.638 + 0.769i)5-s + (0.361 − 0.932i)6-s + (−0.595 − 0.803i)7-s + (0.539 − 0.841i)8-s + (−0.772 + 0.635i)9-s + (0.986 − 0.163i)10-s + (0.374 + 0.927i)11-s + (−0.883 + 0.468i)12-s + (−0.682 − 0.731i)13-s + (−0.0759 + 0.997i)14-s + (−0.939 − 0.341i)15-s + (−0.959 + 0.283i)16-s + (−0.198 + 0.980i)17-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (0.337 + 0.941i)3-s + (0.143 + 0.989i)4-s + (−0.638 + 0.769i)5-s + (0.361 − 0.932i)6-s + (−0.595 − 0.803i)7-s + (0.539 − 0.841i)8-s + (−0.772 + 0.635i)9-s + (0.986 − 0.163i)10-s + (0.374 + 0.927i)11-s + (−0.883 + 0.468i)12-s + (−0.682 − 0.731i)13-s + (−0.0759 + 0.997i)14-s + (−0.939 − 0.341i)15-s + (−0.959 + 0.283i)16-s + (−0.198 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4007\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(430.611\)
Root analytic conductor: \(430.611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4007} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4007,\ (1:\ ),\ -0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1296804311 + 0.3030791101i\)
\(L(\frac12)\) \(\approx\) \(-0.1296804311 + 0.3030791101i\)
\(L(1)\) \(\approx\) \(0.5849963111 + 0.1684541553i\)
\(L(1)\) \(\approx\) \(0.5849963111 + 0.1684541553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4007 \( 1 \)
good2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + (0.337 + 0.941i)T \)
5 \( 1 + (-0.638 + 0.769i)T \)
7 \( 1 + (-0.595 - 0.803i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (-0.682 - 0.731i)T \)
17 \( 1 + (-0.198 + 0.980i)T \)
19 \( 1 + (0.974 - 0.222i)T \)
23 \( 1 + (-0.661 - 0.750i)T \)
29 \( 1 + (-0.865 + 0.501i)T \)
31 \( 1 + (0.981 + 0.191i)T \)
37 \( 1 + (0.712 - 0.701i)T \)
41 \( 1 + (-0.177 + 0.984i)T \)
43 \( 1 + (0.750 + 0.660i)T \)
47 \( 1 + (-0.189 + 0.981i)T \)
53 \( 1 + (-0.576 - 0.817i)T \)
59 \( 1 + (0.593 - 0.804i)T \)
61 \( 1 + (0.977 - 0.208i)T \)
67 \( 1 + (0.577 - 0.816i)T \)
71 \( 1 + (0.966 + 0.257i)T \)
73 \( 1 + (-0.258 + 0.966i)T \)
79 \( 1 + (-0.456 + 0.889i)T \)
83 \( 1 + (-0.995 + 0.0908i)T \)
89 \( 1 + (-0.651 + 0.758i)T \)
97 \( 1 + (-0.773 + 0.634i)T \)
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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

−17.97174322427756361220948300445, −17.11786664841680656733441157825, −16.59810065453909492717162942094, −15.881139628543190548622246175296, −15.408578205150894221786745113876, −14.50756989202758622899091933820, −13.75681669473880264417430374021, −13.34178707161351032635435335401, −12.14215221453317459682998392372, −11.77741424778121659512472469331, −11.29921875358808282157163767223, −9.67833917339404436630088506704, −9.46246703810674703193779232702, −8.64477551961805885303100992062, −8.22678661514599508566189972882, −7.29756295378964647636670224296, −6.96570470468676060437223063221, −5.762226284278690596097695455107, −5.64570963382283767196691784794, −4.4250013872710791630020022677, −3.33633543834767756489681134349, −2.442045192220567763578018500508, −1.58719992907120013466871984794, −0.65679259300295118088135356002, −0.09905958015041531897688648580, 0.97100101141678307200002548098, 2.30032198036383971030993684232, 2.8712366234327794677932604622, 3.703498009412852988805854029143, 4.08505989089148927518699851189, 4.919328107935876715972957220260, 6.32570845481211600909085267792, 7.03063906921135759095466101874, 7.86456048032778958439122425625, 8.15971656434865017048223876719, 9.44778873470074633564602842753, 9.78099292732656077874048560248, 10.34999739132929574422712594783, 10.984389330151158886418404660471, 11.55907065851600750304860729340, 12.55073611908430397763747501488, 13.006582472555678986313874510746, 14.24417024652568199406317976685, 14.60188564382723807954023257258, 15.5985479454646727010578137632, 15.956198376548874221903916396340, 16.78564797915677463124685330817, 17.4255523280014185607580356220, 17.98891699992932546875134329379, 18.97265638815075751076626866399

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