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Properties

Label 1-6003-6003.1163-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.373 + 0.927i$
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  + (−0.737 − 0.675i)2-s + (0.0882 + 0.996i)4-s + (−0.802 + 0.596i)5-s + (0.810 − 0.585i)7-s + (0.607 − 0.794i)8-s + (0.994 + 0.101i)10-s + (−0.628 + 0.777i)11-s + (−0.963 + 0.268i)13-s + (−0.993 − 0.115i)14-s + (−0.984 + 0.175i)16-s + (−0.281 − 0.959i)17-s + (0.574 + 0.818i)19-s + (−0.665 − 0.746i)20-s + (0.988 − 0.149i)22-s + (0.288 − 0.957i)25-s + (0.891 + 0.452i)26-s + ⋯
L(s)  = 1  + (−0.737 − 0.675i)2-s + (0.0882 + 0.996i)4-s + (−0.802 + 0.596i)5-s + (0.810 − 0.585i)7-s + (0.607 − 0.794i)8-s + (0.994 + 0.101i)10-s + (−0.628 + 0.777i)11-s + (−0.963 + 0.268i)13-s + (−0.993 − 0.115i)14-s + (−0.984 + 0.175i)16-s + (−0.281 − 0.959i)17-s + (0.574 + 0.818i)19-s + (−0.665 − 0.746i)20-s + (0.988 − 0.149i)22-s + (0.288 − 0.957i)25-s + (0.891 + 0.452i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.373 + 0.927i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.373 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5336634389 + 0.3605435129i\)
\(L(\frac12)\) \(\approx\) \(0.5336634389 + 0.3605435129i\)
\(L(1)\) \(\approx\) \(0.6200906527 - 0.05644941693i\)
\(L(1)\) \(\approx\) \(0.6200906527 - 0.05644941693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.737 - 0.675i)T \)
5 \( 1 + (-0.802 + 0.596i)T \)
7 \( 1 + (0.810 - 0.585i)T \)
11 \( 1 + (-0.628 + 0.777i)T \)
13 \( 1 + (-0.963 + 0.268i)T \)
17 \( 1 + (-0.281 - 0.959i)T \)
19 \( 1 + (0.574 + 0.818i)T \)
31 \( 1 + (-0.482 + 0.876i)T \)
37 \( 1 + (0.806 - 0.591i)T \)
41 \( 1 + (-0.189 - 0.981i)T \)
43 \( 1 + (0.482 + 0.876i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (0.992 + 0.122i)T \)
59 \( 1 + (-0.0475 - 0.998i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.777 + 0.628i)T \)
71 \( 1 + (0.996 - 0.0815i)T \)
73 \( 1 + (-0.670 + 0.742i)T \)
79 \( 1 + (-0.764 - 0.644i)T \)
83 \( 1 + (-0.760 + 0.649i)T \)
89 \( 1 + (0.639 - 0.768i)T \)
97 \( 1 + (-0.844 + 0.534i)T \)
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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

−17.419613953574569576071390282341, −16.97995150774896966885988564974, −16.33681795640914570709493166267, −15.58034409960819119460090656971, −15.1150005125930533265763291281, −14.74647073486272550875345600531, −13.693896317514161419446551625931, −13.1032863681501654473698418327, −12.153497180758519831203814585400, −11.53523891746272005035397253427, −10.96037644488883499999259204154, −10.278934769057486894737828877730, −9.29883838993332657862700559127, −8.82193888243814934563075557331, −8.11113806914201466077841400171, −7.75740560815610652820180004638, −7.07376545412905487435379069314, −6.00068082793908205668266289716, −5.409052678047089864409649683081, −4.84508937056265847675754001791, −4.132171398110804219765546290919, −2.89688619984959926045690229658, −2.1319205259792999198742237707, −1.15592987398463477152650432374, −0.29692661802924036385535776642, 0.77949195605426713668806626445, 1.79112837383597581146429647991, 2.530004273766366005477379210156, 3.166920128608979309824739120009, 4.19788005352715161050130644030, 4.515069757816128612091256816980, 5.480452551934807315368632822302, 6.895702892958361421977048161734, 7.37966573536265878735341640917, 7.63126656098049420481327415303, 8.43061115775091608595420738222, 9.31694346230876622034742758009, 10.00291979272682692674857680948, 10.58440608002403649642126592209, 11.15128186322063379263683928752, 11.8389019575875002657451937292, 12.27290421974104805570262701217, 13.02909553064663643884900052567, 14.06783869663113620528028209207, 14.43733045192701127611244147312, 15.33519274400237998068514329922, 16.01798924078748076846061984785, 16.58525137048966744543431275708, 17.42377095697491195583720699500, 17.9882223602967807058225643272

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