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Properties

Label 1-6031-6031.1558-r0-0-0
Degree $1$
Conductor $6031$
Sign $0.901 - 0.433i$
Analytic cond. $28.0078$
Root an. cond. $28.0078$
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  + (0.431 − 0.902i)2-s + (0.0193 − 0.999i)3-s + (−0.627 − 0.778i)4-s + (0.0581 − 0.998i)5-s + (−0.893 − 0.448i)6-s + (−0.565 − 0.824i)7-s + (−0.973 + 0.230i)8-s + (−0.999 − 0.0387i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (−0.790 + 0.612i)12-s + (−0.597 − 0.802i)13-s + (−0.987 + 0.154i)14-s + (−0.996 − 0.0774i)15-s + (−0.211 + 0.977i)16-s + (0.993 + 0.116i)17-s + ⋯
L(s)  = 1  + (0.431 − 0.902i)2-s + (0.0193 − 0.999i)3-s + (−0.627 − 0.778i)4-s + (0.0581 − 0.998i)5-s + (−0.893 − 0.448i)6-s + (−0.565 − 0.824i)7-s + (−0.973 + 0.230i)8-s + (−0.999 − 0.0387i)9-s + (−0.875 − 0.483i)10-s + (−0.627 − 0.778i)11-s + (−0.790 + 0.612i)12-s + (−0.597 − 0.802i)13-s + (−0.987 + 0.154i)14-s + (−0.996 − 0.0774i)15-s + (−0.211 + 0.977i)16-s + (0.993 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6031\)    =    \(37 \cdot 163\)
Sign: $0.901 - 0.433i$
Analytic conductor: \(28.0078\)
Root analytic conductor: \(28.0078\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6031} (1558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6031,\ (0:\ ),\ 0.901 - 0.433i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6949047636 + 0.1584054654i\)
\(L(\frac12)\) \(\approx\) \(-0.6949047636 + 0.1584054654i\)
\(L(1)\) \(\approx\) \(0.1529320792 - 0.8648299907i\)
\(L(1)\) \(\approx\) \(0.1529320792 - 0.8648299907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
163 \( 1 \)
good2 \( 1 + (0.431 - 0.902i)T \)
3 \( 1 + (0.0193 - 0.999i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (-0.565 - 0.824i)T \)
11 \( 1 + (-0.627 - 0.778i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.993 + 0.116i)T \)
19 \( 1 + (-0.713 - 0.700i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.910 + 0.413i)T \)
31 \( 1 + (-0.396 + 0.918i)T \)
41 \( 1 + (-0.875 - 0.483i)T \)
43 \( 1 + (-0.856 - 0.516i)T \)
47 \( 1 + (0.0968 - 0.995i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (0.856 - 0.516i)T \)
71 \( 1 + (0.657 - 0.753i)T \)
73 \( 1 + (0.657 - 0.753i)T \)
79 \( 1 + (0.999 - 0.0387i)T \)
83 \( 1 + (-0.910 + 0.413i)T \)
89 \( 1 + (-0.0193 - 0.999i)T \)
97 \( 1 + (0.740 - 0.672i)T \)
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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

−18.379583226656313716214630806484, −17.49862834186057472303983336286, −16.994266260322446240487066475829, −16.24606672372411596202780292284, −15.61840583973352934852249495332, −15.24087321453977992254418916891, −14.51446509331876312882662059604, −14.29205544152200983747022180314, −13.32479223734524958652690005903, −12.51659628423515965954415811793, −11.85301004536444976138545668141, −11.28201978290699652088572348591, −10.04836580286267740500825336395, −9.82132595875062446107907045382, −9.270927957824032811779635885127, −8.13932274050672228146996218998, −7.804292010859101630362662446367, −6.72702154871247493401074709507, −6.294663752833915401532839244712, −5.47431983903837839305273956169, −4.99140737639858603392244593186, −4.05718734924464000693170146657, −3.46807870868913183967113931958, −2.741389362858628479026872812473, −2.10241328657307412870150610248, 0.2191655492119404665130298130, 0.66128999065042501026235030884, 1.46612799583470492745258165062, 2.3844891963722821217173320248, 3.11734683894780880687042010061, 3.70376006349236350903886724947, 4.87567511918270308245964838593, 5.226875047577212520982912476022, 6.07205432214142839053827432090, 6.75768689343488139036054362961, 7.70231233472632831313935536417, 8.4781041339861860884294808318, 8.8232974138457989414585697526, 9.96747317298068949571912038705, 10.40463574853482993030823580754, 11.09142225855089447115511544592, 12.08448403148439556175355760067, 12.50064823657309180601389970987, 12.89765685163383582511216491676, 13.6217885340396892293261392361, 13.92862218156358021184971182558, 14.795979815429711080277312884531, 15.66406920184648158182826598557, 16.62120427537575489279692169805, 17.00463231434342159334851910008

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