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Properties

Label 1-6003-6003.1060-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.967 + 0.251i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.998 − 0.0543i)2-s + (0.994 − 0.108i)4-s + (−0.912 − 0.409i)5-s + (0.777 + 0.628i)7-s + (0.986 − 0.162i)8-s + (−0.933 − 0.359i)10-s + (0.938 + 0.346i)11-s + (0.440 + 0.897i)13-s + (0.810 + 0.585i)14-s + (0.976 − 0.215i)16-s + (−0.654 − 0.755i)17-s + (−0.591 − 0.806i)19-s + (−0.951 − 0.307i)20-s + (0.955 + 0.294i)22-s + (0.665 + 0.746i)25-s + (0.488 + 0.872i)26-s + ⋯
L(s)  = 1  + (0.998 − 0.0543i)2-s + (0.994 − 0.108i)4-s + (−0.912 − 0.409i)5-s + (0.777 + 0.628i)7-s + (0.986 − 0.162i)8-s + (−0.933 − 0.359i)10-s + (0.938 + 0.346i)11-s + (0.440 + 0.897i)13-s + (0.810 + 0.585i)14-s + (0.976 − 0.215i)16-s + (−0.654 − 0.755i)17-s + (−0.591 − 0.806i)19-s + (−0.951 − 0.307i)20-s + (0.955 + 0.294i)22-s + (0.665 + 0.746i)25-s + (0.488 + 0.872i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.967 + 0.251i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1060, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.967 + 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.936742875 + 0.5024645080i\)
\(L(\frac12)\) \(\approx\) \(3.936742875 + 0.5024645080i\)
\(L(1)\) \(\approx\) \(2.064728295 + 0.03717888186i\)
\(L(1)\) \(\approx\) \(2.064728295 + 0.03717888186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.998 - 0.0543i)T \)
5 \( 1 + (-0.912 - 0.409i)T \)
7 \( 1 + (0.777 + 0.628i)T \)
11 \( 1 + (0.938 + 0.346i)T \)
13 \( 1 + (0.440 + 0.897i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.591 - 0.806i)T \)
31 \( 1 + (0.990 + 0.135i)T \)
37 \( 1 + (-0.523 + 0.852i)T \)
41 \( 1 + (-0.888 + 0.458i)T \)
43 \( 1 + (0.990 - 0.135i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.685 - 0.728i)T \)
59 \( 1 + (-0.786 - 0.618i)T \)
61 \( 1 + (0.894 - 0.446i)T \)
67 \( 1 + (0.938 - 0.346i)T \)
71 \( 1 + (0.0203 - 0.999i)T \)
73 \( 1 + (0.182 + 0.983i)T \)
79 \( 1 + (-0.675 + 0.737i)T \)
83 \( 1 + (0.644 - 0.764i)T \)
89 \( 1 + (-0.818 - 0.574i)T \)
97 \( 1 + (0.963 + 0.268i)T \)
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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

−17.42205341482449084915227609754, −17.017612351617911637204880842194, −16.2080694517245649933283098826, −15.519020260648514734741171143196, −14.96536990378577697563221067491, −14.49356934553989930737503809162, −13.81912642408019381445603340480, −13.2044550945529756549033010611, −12.27990986483342319447073705181, −11.92321474522993711064357155415, −11.03089811551820333357378819121, −10.76357565567121732700227660035, −10.115229296139486040145971263073, −8.57400132175053511802350255184, −8.29667819391857555866084332444, −7.46016387947261527302487136328, −6.867567016461824653568915373927, −6.1580895499343475658528028435, −5.46205122760835998592821350779, −4.45534060712540004066339541936, −3.94369446329901874684743905378, −3.58392805960671362215675982464, −2.57426442927657897783370129708, −1.67006910379562707744740260801, −0.80355346015816097634936669391, 0.975789109232859803412215870217, 1.78209908356651061545785549822, 2.52301177802947282424842725175, 3.43971602037064079674820128548, 4.28075135885342381064620459268, 4.63984352310795734838176377444, 5.20863962515372930373785227895, 6.374683733805061187864638945259, 6.74888781191382743670520783587, 7.5372209626614845346882483951, 8.4184471023465524347457133613, 8.896458898259817269838697603137, 9.74581300662278112883929992110, 10.95142377038146028336160521328, 11.386803995312774565559479815708, 11.8013984283803924968808583458, 12.36518225746765906647941729409, 13.113760748183469998369581791411, 13.93747073761065832708042516613, 14.38680903592926214655820234872, 15.24827198703778037723259129217, 15.53729123093189137513064618894, 16.18344429163921672658421750845, 17.02171959451027100477516109151, 17.51451513009716775700165955617

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