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Properties

Label 1-6004-6004.1639-r0-0-0
Degree $1$
Conductor $6004$
Sign $0.944 - 0.327i$
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  + (−0.956 + 0.291i)3-s + (0.653 − 0.757i)5-s + (0.799 − 0.600i)7-s + (0.830 − 0.556i)9-s + (0.354 + 0.935i)11-s + (0.964 − 0.265i)13-s + (−0.404 + 0.914i)15-s + (0.0938 + 0.995i)17-s + (−0.589 + 0.807i)21-s + (−0.766 − 0.642i)23-s + (−0.147 − 0.989i)25-s + (−0.632 + 0.774i)27-s + (−0.994 + 0.107i)29-s + (0.845 + 0.534i)31-s + (−0.611 − 0.791i)33-s + ⋯
L(s)  = 1  + (−0.956 + 0.291i)3-s + (0.653 − 0.757i)5-s + (0.799 − 0.600i)7-s + (0.830 − 0.556i)9-s + (0.354 + 0.935i)11-s + (0.964 − 0.265i)13-s + (−0.404 + 0.914i)15-s + (0.0938 + 0.995i)17-s + (−0.589 + 0.807i)21-s + (−0.766 − 0.642i)23-s + (−0.147 − 0.989i)25-s + (−0.632 + 0.774i)27-s + (−0.994 + 0.107i)29-s + (0.845 + 0.534i)31-s + (−0.611 − 0.791i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (1639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ 0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.797246492 - 0.3023370695i\)
\(L(\frac12)\) \(\approx\) \(1.797246492 - 0.3023370695i\)
\(L(1)\) \(\approx\) \(1.071120727 - 0.07609461146i\)
\(L(1)\) \(\approx\) \(1.071120727 - 0.07609461146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.956 + 0.291i)T \)
5 \( 1 + (0.653 - 0.757i)T \)
7 \( 1 + (0.799 - 0.600i)T \)
11 \( 1 + (0.354 + 0.935i)T \)
13 \( 1 + (0.964 - 0.265i)T \)
17 \( 1 + (0.0938 + 0.995i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.994 + 0.107i)T \)
31 \( 1 + (0.845 + 0.534i)T \)
37 \( 1 + (-0.948 - 0.316i)T \)
41 \( 1 + (-0.653 + 0.757i)T \)
43 \( 1 + (0.329 + 0.944i)T \)
47 \( 1 + (0.930 + 0.367i)T \)
53 \( 1 + (-0.730 + 0.682i)T \)
59 \( 1 + (-0.252 - 0.967i)T \)
61 \( 1 + (-0.523 - 0.852i)T \)
67 \( 1 + (-0.379 - 0.925i)T \)
71 \( 1 + (0.998 - 0.0536i)T \)
73 \( 1 + (0.964 + 0.265i)T \)
83 \( 1 + (0.996 + 0.0804i)T \)
89 \( 1 + (-0.452 - 0.891i)T \)
97 \( 1 + (-0.147 + 0.989i)T \)
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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

−17.958268510302345089577848062951, −17.09292592469502087759029401789, −16.67498361051627955821421944638, −15.65556941693113278964598892129, −15.39170172055571057979226638848, −14.25531162292089826617445872009, −13.71949914745092079457793032249, −13.435268746568540574586698400021, −12.206901580815661115402980316939, −11.70690081923634054286306555298, −11.204314929643139178690614162260, −10.68634052253831480193927720740, −9.901004791312366356375423019867, −9.10425761662947615404341253998, −8.40159205884275315771522566852, −7.49727668120867385227595646457, −6.90121481417587378667223063226, −6.02304729794110706538222256205, −5.74176101774785338542572452869, −5.09324200095609569445242739375, −4.061882419163147515092219356624, −3.28580754402451777163174657349, −2.214257226859737204599091338593, −1.68737916761352384470288668273, −0.767856447796189234101256090686, 0.718569414657292713969359060436, 1.55498912907297678458469670725, 1.89388882733693696979769366485, 3.49092184371720099494584566775, 4.23795015207829910088625287828, 4.714546039764061928710084939544, 5.38148145600282613135511100624, 6.21124026353140957385317022712, 6.58797182584878291752580557749, 7.696735211362695719107253271164, 8.268654324822288307843930848483, 9.15567115225578981351844279196, 9.80080198898070612539083042706, 10.54132747089428683321902861163, 10.876733495084415904084610845796, 11.78837853807029259638399362030, 12.4783326999381865891742962436, 12.852257016536117250227871125, 13.76695371956223607819209982613, 14.338625426517901278995576421, 15.22189132940359162975875780754, 15.778205726788548067745577117422, 16.60627130594267872262016927381, 17.09881880208289509273015170228, 17.52416747453715492452006545073

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