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Properties

Label 1-6004-6004.1499-r0-0-0
Degree $1$
Conductor $6004$
Sign $-0.844 - 0.534i$
Analytic cond. $27.8824$
Root an. cond. $27.8824$
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  + (−0.147 + 0.989i)3-s + (0.909 + 0.416i)5-s + (0.948 + 0.316i)7-s + (−0.956 − 0.291i)9-s + (−0.568 − 0.822i)11-s + (−0.991 − 0.133i)13-s + (−0.545 + 0.837i)15-s + (0.673 + 0.739i)17-s + (−0.452 + 0.891i)21-s + (0.939 − 0.342i)23-s + (0.653 + 0.757i)25-s + (0.428 − 0.903i)27-s + (−0.998 − 0.0536i)29-s + (−0.278 − 0.960i)31-s + (0.897 − 0.440i)33-s + ⋯
L(s)  = 1  + (−0.147 + 0.989i)3-s + (0.909 + 0.416i)5-s + (0.948 + 0.316i)7-s + (−0.956 − 0.291i)9-s + (−0.568 − 0.822i)11-s + (−0.991 − 0.133i)13-s + (−0.545 + 0.837i)15-s + (0.673 + 0.739i)17-s + (−0.452 + 0.891i)21-s + (0.939 − 0.342i)23-s + (0.653 + 0.757i)25-s + (0.428 − 0.903i)27-s + (−0.998 − 0.0536i)29-s + (−0.278 − 0.960i)31-s + (0.897 − 0.440i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $-0.844 - 0.534i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ -0.844 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1567170091 + 0.5404165759i\)
\(L(\frac12)\) \(\approx\) \(-0.1567170091 + 0.5404165759i\)
\(L(1)\) \(\approx\) \(0.9084416811 + 0.4141163701i\)
\(L(1)\) \(\approx\) \(0.9084416811 + 0.4141163701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.147 + 0.989i)T \)
5 \( 1 + (0.909 + 0.416i)T \)
7 \( 1 + (0.948 + 0.316i)T \)
11 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (-0.991 - 0.133i)T \)
17 \( 1 + (0.673 + 0.739i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.998 - 0.0536i)T \)
31 \( 1 + (-0.278 - 0.960i)T \)
37 \( 1 + (-0.987 + 0.160i)T \)
41 \( 1 + (-0.909 - 0.416i)T \)
43 \( 1 + (-0.815 + 0.579i)T \)
47 \( 1 + (-0.982 + 0.186i)T \)
53 \( 1 + (-0.930 - 0.367i)T \)
59 \( 1 + (0.611 + 0.791i)T \)
61 \( 1 + (0.872 - 0.488i)T \)
67 \( 1 + (-0.830 + 0.556i)T \)
71 \( 1 + (-0.999 - 0.0268i)T \)
73 \( 1 + (-0.991 + 0.133i)T \)
83 \( 1 + (0.0402 + 0.999i)T \)
89 \( 1 + (0.523 + 0.852i)T \)
97 \( 1 + (0.653 - 0.757i)T \)
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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

−17.54524268057759057190710263706, −16.8878341484072969972224275814, −16.3158732917165027135206996678, −15.12737095592268832743447689406, −14.50594522370328388319689516587, −14.08180583394842044844087162561, −13.218453767928688380200342264359, −12.91050096097722340215053666850, −12.01446627077081660500580809305, −11.658980684952126463342686869993, −10.642653692463381739898665474588, −10.09069526562376971693827149320, −9.26781179284840186559168717867, −8.59555548524537023547228223047, −7.76324010776929942036980576769, −7.22246918996611562507277775371, −6.75207666319707810107827785202, −5.61274871000057317227662406896, −5.01496781192514006775410754049, −4.85322943724784177726701510570, −3.32627149389564384244469292911, −2.5090370428601485696243612613, −1.70253894197455238371670760602, −1.41884529471908452838740113517, −0.12554695095202768195521802024, 1.33975911560029936730203929572, 2.228349938537517143873253448, 2.93877156350824465396939739341, 3.6038080314886625642815758097, 4.656636994793589864259582058076, 5.33751077980904347632127857953, 5.582925047043660905415290820337, 6.45614776920969831420148260398, 7.42429577014966260422633883107, 8.25015010592462704904200969944, 8.7988125688208069923440284588, 9.62794654718190043228427919779, 10.14900682409767821075879094874, 10.79411117251849102039628874646, 11.28354510344557701709596506618, 12.00257079834434199006595545981, 12.95471665000378574823819864340, 13.59649050578761099405973787937, 14.50092333858176636344210107484, 14.79348306912258330148739997100, 15.22352907745266893288297649215, 16.29916215580562275108764389480, 16.83230209338028484707153603574, 17.37972962690188411858712615457, 17.882548258225313101644503641881

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