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Properties

Label 1-4029-4029.2123-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.303 - 0.952i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.663 + 0.748i)2-s + (−0.120 + 0.992i)4-s + (0.517 − 0.855i)5-s + (−0.983 + 0.180i)7-s + (−0.822 + 0.568i)8-s + (0.983 − 0.180i)10-s + (−0.517 − 0.855i)11-s + (−0.120 + 0.992i)13-s + (−0.787 − 0.616i)14-s + (−0.970 − 0.239i)16-s + (−0.464 + 0.885i)19-s + (0.787 + 0.616i)20-s + (0.297 − 0.954i)22-s + (0.707 + 0.707i)23-s + (−0.464 − 0.885i)25-s + (−0.822 + 0.568i)26-s + ⋯
L(s)  = 1  + (0.663 + 0.748i)2-s + (−0.120 + 0.992i)4-s + (0.517 − 0.855i)5-s + (−0.983 + 0.180i)7-s + (−0.822 + 0.568i)8-s + (0.983 − 0.180i)10-s + (−0.517 − 0.855i)11-s + (−0.120 + 0.992i)13-s + (−0.787 − 0.616i)14-s + (−0.970 − 0.239i)16-s + (−0.464 + 0.885i)19-s + (0.787 + 0.616i)20-s + (0.297 − 0.954i)22-s + (0.707 + 0.707i)23-s + (−0.464 − 0.885i)25-s + (−0.822 + 0.568i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.303 - 0.952i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6005157374 - 0.4391448250i\)
\(L(\frac12)\) \(\approx\) \(0.6005157374 - 0.4391448250i\)
\(L(1)\) \(\approx\) \(1.065278390 + 0.3539929026i\)
\(L(1)\) \(\approx\) \(1.065278390 + 0.3539929026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.663 + 0.748i)T \)
5 \( 1 + (0.517 - 0.855i)T \)
7 \( 1 + (-0.983 + 0.180i)T \)
11 \( 1 + (-0.517 - 0.855i)T \)
13 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 + (-0.464 + 0.885i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.410 + 0.911i)T \)
31 \( 1 + (0.0603 - 0.998i)T \)
37 \( 1 + (0.954 - 0.297i)T \)
41 \( 1 + (-0.855 - 0.517i)T \)
43 \( 1 + (-0.239 - 0.970i)T \)
47 \( 1 + (-0.885 + 0.464i)T \)
53 \( 1 + (-0.822 + 0.568i)T \)
59 \( 1 + (0.992 - 0.120i)T \)
61 \( 1 + (-0.297 - 0.954i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (-0.983 - 0.180i)T \)
73 \( 1 + (0.787 + 0.616i)T \)
83 \( 1 + (-0.992 - 0.120i)T \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (0.954 - 0.297i)T \)
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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

−18.75389255112811353054470439798, −17.92885647009873975113906540724, −17.616896730541775253676230121104, −16.47285329007546330672518831077, −15.59441848407541466123503021, −14.99346922012684931556375112736, −14.636874223192640288120586515184, −13.479995210622667068455422818460, −13.195513833785419010546045317679, −12.657132086619351217489221132661, −11.77338829664962503075153026153, −10.93566353417807984845891888682, −10.3029046402446383232947342718, −9.95304958790528589292948885112, −9.275668769730259005490365272491, −8.18367560582301435920058602341, −7.02440165114801050575090209874, −6.6227232398019004487334940436, −5.887277558365400628128800383264, −5.02518610699816125748655948000, −4.36675644757742440571394966864, −3.22394927893301024523295956857, −2.84955829155294612971071750829, −2.23439495938172934903907737713, −1.06658321392730718621722401998, 0.1637186493151113074443629295, 1.60621846614741262146250641765, 2.59399344309018157640454011893, 3.408987830804759969173977578911, 4.140101723070753925935775219505, 5.003290766509279761307350955847, 5.672254922970391640145090170298, 6.22684129690622487305380382991, 6.89184800045274195779515443382, 7.85113782120325774288959972869, 8.57000503830139157497110029912, 9.17804556791220005331259125060, 9.76846028335948477253011008948, 10.83043279906568604106210032321, 11.77983314738863346077623096781, 12.38433900230848367438895026526, 13.09342942849845232412312505501, 13.45636888886524237965496518890, 14.16268159582023759880590124536, 14.9112912051282074808537665410, 15.92337718347819905542478090575, 16.12589537928529983192371217370, 16.93635697062400470182260509572, 17.16428137918074573175695519366, 18.378817242591760981786849078405

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