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Properties

Label 1-6025-6025.191-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.573 - 0.818i$
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  + (0.743 + 0.669i)2-s + (0.207 − 0.978i)3-s + (0.104 + 0.994i)4-s + (0.809 − 0.587i)6-s + (−0.933 + 0.358i)7-s + (−0.587 + 0.809i)8-s + (−0.913 − 0.406i)9-s + (0.0523 + 0.998i)11-s + (0.994 + 0.104i)12-s + (−0.933 − 0.358i)13-s + (−0.933 − 0.358i)14-s + (−0.978 + 0.207i)16-s + (−0.707 − 0.707i)17-s + (−0.406 − 0.913i)18-s + (0.629 + 0.777i)19-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.207 − 0.978i)3-s + (0.104 + 0.994i)4-s + (0.809 − 0.587i)6-s + (−0.933 + 0.358i)7-s + (−0.587 + 0.809i)8-s + (−0.913 − 0.406i)9-s + (0.0523 + 0.998i)11-s + (0.994 + 0.104i)12-s + (−0.933 − 0.358i)13-s + (−0.933 − 0.358i)14-s + (−0.978 + 0.207i)16-s + (−0.707 − 0.707i)17-s + (−0.406 − 0.913i)18-s + (0.629 + 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.573 - 0.818i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.573 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7792869540 - 0.4054317835i\)
\(L(\frac12)\) \(\approx\) \(0.7792869540 - 0.4054317835i\)
\(L(1)\) \(\approx\) \(1.084732314 + 0.2496433474i\)
\(L(1)\) \(\approx\) \(1.084732314 + 0.2496433474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (-0.933 + 0.358i)T \)
11 \( 1 + (0.0523 + 0.998i)T \)
13 \( 1 + (-0.933 - 0.358i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.629 + 0.777i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (-0.156 - 0.987i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 - iT \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.258 + 0.965i)T \)
73 \( 1 + (0.156 - 0.987i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.998 + 0.0523i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−17.814736089053583812635271635210, −16.821352509851162452645193357434, −16.44660802683076513922494578301, −15.70603286847958055678369904846, −15.142323929693236345163557099591, −14.47658006449624261830123706072, −13.84283270495581953795546635363, −13.2741091837363683198672188024, −12.67163125374008456824228344809, −11.70177582687597127615734232180, −11.19664417312637945322218298917, −10.62816619392452301335143452738, −9.79406563912435361914733232796, −9.50905990476133851419094498066, −8.78201042935108240422145164731, −7.79749776484885662523605282314, −6.77149399638020914516709695134, −6.05949494269449977271834459097, −5.54484431683088634860352519229, −4.52397968827002012823864058565, −4.1418175113873311244011095608, −3.410777679402252762449418292643, −2.72393381526873853655443346460, −2.15592200471227105526988742735, −0.73776244446778476619415258722, 0.19853079585444988997807016676, 1.83731400368134277991078946141, 2.31929044702353706244943114182, 3.25448364649014470992931235713, 3.68390059829363593371815803608, 4.98465806092583939803516144503, 5.34675182534089175104008897296, 6.28530501256259928405054443744, 6.81621049761349589284270660969, 7.39157057249096220487939765057, 7.8877771250365279219584937459, 8.821282735344820217786662315995, 9.46440996431238698420800080575, 10.147482215980251246618555724193, 11.535499341796758852485856118865, 11.88188455183849837401324693298, 12.69753844292280414262919977009, 12.83807563124768734242695888443, 13.71855292987755862217493586710, 14.24509486951647064798793646024, 14.99471109391289691143005028947, 15.47234516765799042227115191649, 16.26722286041642372759711948307, 16.90048402066068204295708258294, 17.6235350435576246858371545308

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