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Properties

Label 1-5577-5577.230-r0-0-0
Degree $1$
Conductor $5577$
Sign $-0.163 - 0.986i$
Analytic cond. $25.8994$
Root an. cond. $25.8994$
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  + (0.799 − 0.600i)2-s + (0.278 − 0.960i)4-s + (−0.885 − 0.464i)5-s + (−0.987 − 0.160i)7-s + (−0.354 − 0.935i)8-s + (−0.987 + 0.160i)10-s + (−0.885 + 0.464i)14-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (0.5 − 0.866i)19-s + (−0.692 + 0.721i)20-s + (0.5 + 0.866i)23-s + (0.568 + 0.822i)25-s + (−0.428 + 0.903i)28-s + (0.799 − 0.600i)29-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + (0.278 − 0.960i)4-s + (−0.885 − 0.464i)5-s + (−0.987 − 0.160i)7-s + (−0.354 − 0.935i)8-s + (−0.987 + 0.160i)10-s + (−0.885 + 0.464i)14-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (0.5 − 0.866i)19-s + (−0.692 + 0.721i)20-s + (0.5 + 0.866i)23-s + (0.568 + 0.822i)25-s + (−0.428 + 0.903i)28-s + (0.799 − 0.600i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5577\)    =    \(3 \cdot 11 \cdot 13^{2}\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(25.8994\)
Root analytic conductor: \(25.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5577} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5577,\ (0:\ ),\ -0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429619023 - 1.685995960i\)
\(L(\frac12)\) \(\approx\) \(1.429619023 - 1.685995960i\)
\(L(1)\) \(\approx\) \(1.157405504 - 0.7075693640i\)
\(L(1)\) \(\approx\) \(1.157405504 - 0.7075693640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.799 - 0.600i)T \)
5 \( 1 + (-0.885 - 0.464i)T \)
7 \( 1 + (-0.987 - 0.160i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.799 - 0.600i)T \)
31 \( 1 + (0.568 - 0.822i)T \)
37 \( 1 + (-0.996 - 0.0804i)T \)
41 \( 1 + (-0.200 + 0.979i)T \)
43 \( 1 + (0.996 - 0.0804i)T \)
47 \( 1 + (0.970 + 0.239i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (0.845 - 0.534i)T \)
61 \( 1 + (0.632 + 0.774i)T \)
67 \( 1 + (0.278 + 0.960i)T \)
71 \( 1 + (-0.948 + 0.316i)T \)
73 \( 1 + (-0.120 + 0.992i)T \)
79 \( 1 + (0.970 + 0.239i)T \)
83 \( 1 + (-0.748 + 0.663i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.0402 + 0.999i)T \)
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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

−18.013868440339786383731362021955, −17.1280648520500184900305559152, −16.39812678824311726138676277834, −15.97058626176562531258584631265, −15.534561848273676974463053527685, −14.611643616351880293062906151027, −14.27469024470377750072970853602, −13.51935928628618575119814044658, −12.58582143546598908052376942031, −12.18672956582726827184467641128, −11.79800258000706425002749587725, −10.624187664257397894394001433105, −10.26187383856885748684849687543, −9.0768304978337024610390520089, −8.45923962267810545082921979307, −7.70691673087647053911024796308, −7.02913007303912829167226391227, −6.58718448665020549246761282137, −5.75294677918313904468656172856, −5.06199030031654341102809532159, −4.200044100801348001314232508832, −3.331427830085345333721614430198, −3.2072216820473803273134700653, −2.204201544149430877337658551968, −0.719662035341904994576807020761, 0.69167142345495494597542795791, 1.19820889596140068595786482357, 2.59847985279937444171658586817, 3.07005799172480377499504254837, 3.89062971021582848359829636737, 4.34013505138658928599443501759, 5.32223083369201613278937730714, 5.79747433521683340429024706680, 6.83460277791508029006655152006, 7.29029610378704483699859622912, 8.22772790319968129607116599750, 9.17908677358546821974825703545, 9.70356313082692944626774991393, 10.406713718230277581501299461751, 11.21472055318010731636397222220, 11.85087508294818038463916498899, 12.30131960904811048648008439397, 13.044747582870943865398217776779, 13.50590879971575279763197017069, 14.22962719122386356031654107638, 15.15118213728984586616984276656, 15.61586674171818179181046116480, 16.08943566626015668931654254862, 16.85890156917506379930995437443, 17.643853615570500168108999883128

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