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Properties

Label 1-4009-4009.1033-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.997 + 0.0725i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (−0.772 − 0.635i)2-s + (0.251 + 0.967i)3-s + (0.193 + 0.981i)4-s + (0.251 − 0.967i)5-s + (0.420 − 0.907i)6-s + (−0.772 + 0.635i)7-s + (0.473 − 0.880i)8-s + (−0.873 + 0.486i)9-s + (−0.809 + 0.587i)10-s + (−0.0448 − 0.998i)11-s + (−0.900 + 0.433i)12-s + (−0.925 + 0.379i)13-s + 14-s + 15-s + (−0.925 + 0.379i)16-s + (−0.280 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.772 − 0.635i)2-s + (0.251 + 0.967i)3-s + (0.193 + 0.981i)4-s + (0.251 − 0.967i)5-s + (0.420 − 0.907i)6-s + (−0.772 + 0.635i)7-s + (0.473 − 0.880i)8-s + (−0.873 + 0.486i)9-s + (−0.809 + 0.587i)10-s + (−0.0448 − 0.998i)11-s + (−0.900 + 0.433i)12-s + (−0.925 + 0.379i)13-s + 14-s + 15-s + (−0.925 + 0.379i)16-s + (−0.280 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.997 + 0.0725i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.997 + 0.0725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7071891069 + 0.02568347823i\)
\(L(\frac12)\) \(\approx\) \(0.7071891069 + 0.02568347823i\)
\(L(1)\) \(\approx\) \(0.6359894874 + 0.01337770244i\)
\(L(1)\) \(\approx\) \(0.6359894874 + 0.01337770244i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.772 - 0.635i)T \)
3 \( 1 + (0.251 + 0.967i)T \)
5 \( 1 + (0.251 - 0.967i)T \)
7 \( 1 + (-0.772 + 0.635i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
13 \( 1 + (-0.925 + 0.379i)T \)
17 \( 1 + (-0.280 + 0.959i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (0.791 + 0.611i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (-0.842 - 0.538i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.858 - 0.512i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (-0.337 - 0.941i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.988 - 0.149i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.999 - 0.0299i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.998 + 0.0598i)T \)
97 \( 1 + (0.992 + 0.119i)T \)
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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

−18.31728289325171176497271237978, −17.72249760519188418256600062775, −17.48985614062274673736017673942, −16.57559616671705667674756021234, −15.71826646210723724809441234987, −15.091572038319755209748414965560, −14.34269511600484690028787625348, −13.88389187829365587847685998867, −13.245881920724099082633083424802, −12.19060886693203421442877313247, −11.70418573290150740813951419712, −10.461720254677776750562413661394, −10.18676415400092786722726554112, −9.47556927533199069852496704447, −8.67605246940153340052876740254, −7.557828620495986660520190008861, −7.37076607523588515322839128147, −6.747046088529635793899309300498, −6.19971183008370733495361574108, −5.309941118125547199096226683531, −4.2639275507211037508616852370, −2.94979897304093658158435593512, −2.47607718331645792926923753213, −1.62618772791562734634766292762, −0.47494901036922059481501571456, 0.48976950398479424094629384266, 1.86394425128595863520965562908, 2.48006200165993894447409001650, 3.38097288351185466607287329192, 4.01829781214654542167614969551, 4.828221720085113966043162816754, 5.75812145893723754248554422968, 6.3913635486688513354893661090, 7.72912038949663779729892459189, 8.36903567935449745122587090513, 9.01220027335899002311887880564, 9.350221606771909963314559193008, 10.13450746425209991445730508261, 10.625744764225796967402198901743, 11.67469959872760754961523120072, 12.10568784016488205702237452329, 12.90292451691362056761219950491, 13.55966282694946370861076174762, 14.37931454693138952507095788156, 15.45069876640267444253196811592, 15.967473590247717891812188179972, 16.51494204980604485375261110002, 17.03110656228607045347828819143, 17.66001031647451968902714208753, 18.68882038410245953667159665067

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