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Properties

Label 1-2527-2527.93-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.987 - 0.157i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (0.967 − 0.254i)2-s + (0.989 − 0.146i)3-s + (0.870 − 0.492i)4-s + (−0.999 − 0.0183i)5-s + (0.919 − 0.393i)6-s + (0.716 − 0.697i)8-s + (0.957 − 0.289i)9-s + (−0.971 + 0.236i)10-s + (0.546 + 0.837i)11-s + (0.789 − 0.614i)12-s + (0.315 + 0.948i)13-s + (−0.991 + 0.128i)15-s + (0.515 − 0.856i)16-s + (−0.861 + 0.507i)17-s + (0.851 − 0.523i)18-s + ⋯
L(s)  = 1  + (0.967 − 0.254i)2-s + (0.989 − 0.146i)3-s + (0.870 − 0.492i)4-s + (−0.999 − 0.0183i)5-s + (0.919 − 0.393i)6-s + (0.716 − 0.697i)8-s + (0.957 − 0.289i)9-s + (−0.971 + 0.236i)10-s + (0.546 + 0.837i)11-s + (0.789 − 0.614i)12-s + (0.315 + 0.948i)13-s + (−0.991 + 0.128i)15-s + (0.515 − 0.856i)16-s + (−0.861 + 0.507i)17-s + (0.851 − 0.523i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.987 - 0.157i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.284662347 - 0.3403537623i\)
\(L(\frac12)\) \(\approx\) \(4.284662347 - 0.3403537623i\)
\(L(1)\) \(\approx\) \(2.411535699 - 0.3069326671i\)
\(L(1)\) \(\approx\) \(2.411535699 - 0.3069326671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.967 - 0.254i)T \)
3 \( 1 + (0.989 - 0.146i)T \)
5 \( 1 + (-0.999 - 0.0183i)T \)
11 \( 1 + (0.546 + 0.837i)T \)
13 \( 1 + (0.315 + 0.948i)T \)
17 \( 1 + (-0.861 + 0.507i)T \)
23 \( 1 + (-0.621 + 0.783i)T \)
29 \( 1 + (0.983 - 0.182i)T \)
31 \( 1 + (0.904 - 0.426i)T \)
37 \( 1 + (0.451 - 0.892i)T \)
41 \( 1 + (-0.842 - 0.539i)T \)
43 \( 1 + (0.280 + 0.959i)T \)
47 \( 1 + (-0.119 + 0.992i)T \)
53 \( 1 + (-0.800 - 0.599i)T \)
59 \( 1 + (-0.0459 + 0.998i)T \)
61 \( 1 + (-0.912 - 0.410i)T \)
67 \( 1 + (0.997 - 0.0734i)T \)
71 \( 1 + (0.811 + 0.584i)T \)
73 \( 1 + (0.870 + 0.492i)T \)
79 \( 1 + (0.280 + 0.959i)T \)
83 \( 1 + (-0.754 + 0.656i)T \)
89 \( 1 + (0.870 - 0.492i)T \)
97 \( 1 + (-0.562 + 0.826i)T \)
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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

−19.86702951104235586288013361923, −18.948522062964709455119526424086, −18.19472364594433773457223097094, −17.025678266821008770074515438731, −16.22210093985961589174455488173, −15.60851395740549005513289556223, −15.28737727744238255359264716864, −14.38493799937939711116568725924, −13.82910096975937659736412603519, −13.195941619952109633643881029599, −12.34488568665092927311809353863, −11.736197213575400414932213651912, −10.86075184867845564222519248270, −10.23042813198047366760379393744, −8.88738984107815065193976250922, −8.28388780173867108324514055814, −7.85187746014096219729038819909, −6.80321924031488178846380931847, −6.31720149214264260766222829956, −4.95364850681653151934959843724, −4.46096839725832652828121336021, −3.50146422680302571737023369504, −3.15354700811867510461528723773, −2.258434125400224827198310109694, −0.94660391734559269208580563894, 1.20781978589599229248881146067, 2.00601218342713375868874690814, 2.81272453678573107716973689050, 3.85811851817674101381569796797, 4.15285976452892087120264314745, 4.82976147739512407866359646148, 6.32502646056908412914780844363, 6.78943835184651903934999380163, 7.607796210404925904078992830108, 8.30080005970052947372562414476, 9.27877561021729221279379892078, 9.94862638099124082510736512397, 11.00542915469116554863564396546, 11.64946358110840454185097412862, 12.37051792467748204361009183527, 12.90554636311335584530153532231, 13.87756474171806960190475612272, 14.26577965741270235317106079866, 15.12882070293380666931642912038, 15.579689437679813419817643069972, 16.119210335691774231019611661558, 17.20607661671273431999379956451, 18.26565774157316275374366088710, 19.12238788338934824891253624004, 19.71081365174071518744596033666

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