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Properties

Label 1-6003-6003.344-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.909 - 0.414i$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.826 − 0.563i)5-s + (0.988 + 0.149i)7-s + (0.222 + 0.974i)8-s + (−0.623 − 0.781i)10-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.0747 − 0.997i)14-s + (0.955 − 0.294i)16-s + 17-s + (−0.623 − 0.781i)19-s + (−0.733 + 0.680i)20-s + (−0.365 − 0.930i)22-s + (0.365 − 0.930i)25-s + (−0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.826 − 0.563i)5-s + (0.988 + 0.149i)7-s + (0.222 + 0.974i)8-s + (−0.623 − 0.781i)10-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.0747 − 0.997i)14-s + (0.955 − 0.294i)16-s + 17-s + (−0.623 − 0.781i)19-s + (−0.733 + 0.680i)20-s + (−0.365 − 0.930i)22-s + (0.365 − 0.930i)25-s + (−0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.909 - 0.414i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.909 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4632840199 - 2.133034234i\)
\(L(\frac12)\) \(\approx\) \(0.4632840199 - 2.133034234i\)
\(L(1)\) \(\approx\) \(0.9566685155 - 0.8305294536i\)
\(L(1)\) \(\approx\) \(0.9566685155 - 0.8305294536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (-0.955 - 0.294i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.733 - 0.680i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.365 + 0.930i)T \)
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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

−17.81212156290664378097117238178, −17.32937296356375602524164284910, −16.7277579930957453571848938208, −16.30171292202495329324518134119, −15.000218815961159377762981249041, −14.72454478733401187763273306648, −14.2681726765287716975621694090, −13.82113103906394760836362227843, −12.86752501293590030005257386568, −12.14837696130890387338237793785, −11.407307578770937058922771324054, −10.487398087347518781892537170196, −9.83704653134027474518048062267, −9.40952781945516609413762847717, −8.50558481216625441899002819166, −7.900351522613457927562289875565, −7.11427776024646761574981509064, −6.61820036867707567927810818215, −5.89425261250001583514722584939, −5.2187278696768560998236981450, −4.475831669943384861016764783114, −3.861094983273034054270585926382, −2.77882293146408344849385077225, −1.68496477749240621276533541235, −1.23125196585337378609354748002, 0.58875743918281245217911085532, 1.32067198797612681418686517379, 1.99573435959001141655115699939, 2.71448974485539502964834817950, 3.5691706284072644882993598143, 4.5722830695009138531969515791, 4.94565987438468094710500764294, 5.659925938109367403013827742202, 6.470710682192747487817010085646, 7.64426731227324736282499042180, 8.30993610726697376416916059001, 8.832466442806184107214132160516, 9.593631616844060050483778652897, 10.08766683034097270354168556926, 10.80535322298544419953221264593, 11.63092131956001902581294035280, 12.02705439667409710130706614676, 12.77463943561213578696026091687, 13.38051506954728758455300924238, 14.09825605104526969077956740339, 14.58061677485565057572422457012, 15.20874619410151581355297860277, 16.46699962000567868255607338296, 17.10237785253312324303983951293, 17.49403221322544866093753100765

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