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Properties

Label 1-6003-6003.1093-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: $\chi_{6003} (1093, \cdot )$
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

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

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