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Properties

Label 1-65e2-4225.1334-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.329 + 0.944i$
Analytic cond. $454.039$
Root an. cond. $454.039$
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  + (0.285 − 0.958i)2-s + (0.443 − 0.896i)3-s + (−0.836 − 0.548i)4-s + (−0.732 − 0.681i)6-s + (0.239 − 0.970i)7-s + (−0.764 + 0.644i)8-s + (−0.607 − 0.794i)9-s + (0.794 + 0.607i)11-s + (−0.861 + 0.506i)12-s + (−0.861 − 0.506i)14-s + (0.399 + 0.916i)16-s + (0.644 + 0.764i)17-s + (−0.935 + 0.354i)18-s + (0.587 − 0.809i)19-s + (−0.764 − 0.644i)21-s + (0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (0.285 − 0.958i)2-s + (0.443 − 0.896i)3-s + (−0.836 − 0.548i)4-s + (−0.732 − 0.681i)6-s + (0.239 − 0.970i)7-s + (−0.764 + 0.644i)8-s + (−0.607 − 0.794i)9-s + (0.794 + 0.607i)11-s + (−0.861 + 0.506i)12-s + (−0.861 − 0.506i)14-s + (0.399 + 0.916i)16-s + (0.644 + 0.764i)17-s + (−0.935 + 0.354i)18-s + (0.587 − 0.809i)19-s + (−0.764 − 0.644i)21-s + (0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.329 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9015738238 - 0.6402948602i\)
\(L(\frac12)\) \(\approx\) \(-0.9015738238 - 0.6402948602i\)
\(L(1)\) \(\approx\) \(0.7321024829 - 0.9990491904i\)
\(L(1)\) \(\approx\) \(0.7321024829 - 0.9990491904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.285 - 0.958i)T \)
3 \( 1 + (0.443 - 0.896i)T \)
7 \( 1 + (0.239 - 0.970i)T \)
11 \( 1 + (0.794 + 0.607i)T \)
17 \( 1 + (0.644 + 0.764i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.0241 + 0.999i)T \)
31 \( 1 + (-0.732 - 0.681i)T \)
37 \( 1 + (0.421 - 0.906i)T \)
41 \( 1 + (0.985 - 0.168i)T \)
43 \( 1 + (0.120 + 0.992i)T \)
47 \( 1 + (0.964 - 0.262i)T \)
53 \( 1 + (-0.926 + 0.377i)T \)
59 \( 1 + (0.506 + 0.861i)T \)
61 \( 1 + (-0.527 - 0.849i)T \)
67 \( 1 + (-0.548 - 0.836i)T \)
71 \( 1 + (-0.896 - 0.443i)T \)
73 \( 1 + (-0.794 - 0.607i)T \)
79 \( 1 + (-0.262 - 0.964i)T \)
83 \( 1 + (-0.896 + 0.443i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (-0.976 - 0.215i)T \)
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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

−18.936496814022456183261014818833, −17.94055309903217067203588089777, −17.15566917801736430922064081682, −16.43058611010312605516153420583, −16.085902115618131656948827551575, −15.37974046117006989792908333958, −14.60712339943965424408835786295, −14.32754854222357958303626089494, −13.69600548495441368426550614249, −12.71200269618863388559312875191, −11.93349883870381015888478590250, −11.39393081777763163072090822415, −10.27740157442329078439844319535, −9.510803720697810363620326990637, −9.01632342396334758684396500059, −8.38663801378342764751356811584, −7.800242616324136765411978014091, −6.8632639740198086461857873945, −5.804873875174370881338454103432, −5.58127278788725391198953484899, −4.64828814616515714546630599952, −4.00256586137080427879170096340, −3.154819644144179210907917887039, −2.58282818135983118572608912560, −1.16091546633680736137374509047, 0.13705276247882513939928880926, 1.23953536819373944715096414704, 1.40051658982824844589588256370, 2.44192332623057533699708822961, 3.34572551369902068283762650200, 3.8819532240314976783014367028, 4.6933833467283724519255008319, 5.67145491158811520492489026557, 6.41630644548655189319737544194, 7.48487371038988676061384309152, 7.660763958111755446987765640790, 9.02187324482289923308202026950, 9.22708576706562205882678633871, 10.16890248418895715884490287147, 11.00518981971148793936717496565, 11.527066034888314961645044912346, 12.3191900429003783708386621698, 12.89456120910742627334057102028, 13.443426985928519349500978864858, 14.22606470892117625922574147700, 14.54922875868739710948979122957, 15.26985889006856275863444144688, 16.53783201978399526644465128547, 17.3910117486707484872716395454, 17.71770322067646674515753424472

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