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Properties

Label 1-6025-6025.2611-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.499 + 0.866i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.309 + 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)8-s + 9-s + (0.951 + 0.309i)11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s − i·14-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.309 + 0.951i)18-s + (−0.587 − 0.809i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)8-s + 9-s + (0.951 + 0.309i)11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s − i·14-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.309 + 0.951i)18-s + (−0.587 − 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4810528388 + 0.2778463259i\)
\(L(\frac12)\) \(\approx\) \(0.4810528388 + 0.2778463259i\)
\(L(1)\) \(\approx\) \(0.4686596447 + 0.2481612016i\)
\(L(1)\) \(\approx\) \(0.4686596447 + 0.2481612016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 - T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.68701491122544950620304863826, −16.943362631989543121710663066713, −16.55237428757996417985378834530, −15.98594733477872052270919170461, −14.974810813141856177921834250593, −14.10505108516761938852675780326, −13.42380718064141648803502299171, −12.64771185660734199981990433782, −12.36066583721512994782199013094, −11.65533987546885754553853363322, −10.93327412122213514654545693460, −10.43241619454993355004229512981, −9.64214898154337388168806733164, −9.40610130749940098908425563526, −8.29748055835654724386545929532, −7.551694929376548803699450898419, −6.75493900127236657197605889895, −6.08318728391097802156078205987, −5.34718546788412249357131534822, −4.29262799037262609059370967267, −4.03486510557370773138315066787, −3.03121159450499485296351543821, −2.2732873628775198949093642271, −1.20774704410978332232853316772, −0.505064868633612031851148143527, 0.38943453874180430031986436257, 1.5016817444740528191491547007, 2.32794472380939314628164295704, 3.90343266415128845091834023427, 4.199889553411035367220476952618, 5.051600212525219871306997590610, 5.86058372877965999451168677257, 6.51200078402272121395707791431, 6.74970051296141687614506462294, 7.47643120105694602798815615252, 8.57733464328141160840556260747, 9.149071681032498103797815763134, 9.80631867019036186766972571895, 10.31097795887981478377360090037, 11.17436936870837729928401895483, 12.01649517408991557789490021080, 12.50784626353585535396101282319, 13.28363876590167161262834100954, 13.889571621903584847129565880171, 14.870425351933195059944072915939, 15.288771567814597970072587822975, 16.07147020619359357757351189799, 16.48007574137657994232667286875, 17.22179013432575161302962812221, 17.51018078824489085234026781867

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