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Properties

Label 1-5275-5275.652-r1-0-0
Degree $1$
Conductor $5275$
Sign $0.552 + 0.833i$
Analytic cond. $566.877$
Root an. cond. $566.877$
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  + (0.207 − 0.978i)2-s + (0.743 − 0.669i)3-s + (−0.913 − 0.406i)4-s + (−0.5 − 0.866i)6-s + (0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.104 − 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.978 − 0.207i)14-s + (0.669 + 0.743i)16-s + (−0.866 − 0.5i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.743 − 0.669i)3-s + (−0.913 − 0.406i)4-s + (−0.5 − 0.866i)6-s + (0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.104 − 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.978 − 0.207i)14-s + (0.669 + 0.743i)16-s + (−0.866 − 0.5i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5275\)    =    \(5^{2} \cdot 211\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(566.877\)
Root analytic conductor: \(566.877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5275} (652, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5275,\ (1:\ ),\ 0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9258390421 - 0.4973616172i\)
\(L(\frac12)\) \(\approx\) \(-0.9258390421 - 0.4973616172i\)
\(L(1)\) \(\approx\) \(0.7022752374 - 0.8908165605i\)
\(L(1)\) \(\approx\) \(0.7022752374 - 0.8908165605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
7 \( 1 + (0.406 + 0.913i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.743 - 0.669i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.207 + 0.978i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−18.165291490022461002626976525470, −17.493117671268974272308697706, −16.80087488286721303092846764383, −16.409016792862133992920305440896, −15.552820048446243342101460016441, −14.877094044617784135163775728916, −14.71667190320562669143867334380, −13.74286444006904507107691844728, −13.36632807705993737952753984735, −12.69663055989554985543708987673, −11.69982994167825857815656304813, −10.72058090991934505634592779186, −10.07323559048194639580798285740, −9.57549185777613598556451715736, −8.74977635138641201899115616843, −8.07530825568657576730315531622, −7.39181108696211684082552456015, −7.12700340831094724076353938098, −5.941857003306146027700425359705, −5.03311203220557895692989000252, −4.61485580329764902483435744969, −4.005735708474130853398146683320, −3.19023103890766054536545248062, −2.31637345194863619133186660884, −1.254919048481130297955232353149, 0.14946888204227311350151471549, 0.71362256614476952789423829033, 1.91722693016678258772195298956, 2.41561318463785846840963944884, 2.91907608494554978325615725612, 3.66622236690615479887292904403, 4.94648856359937413971819650109, 5.1305909679501413553717095472, 6.16303441600391705127481301632, 7.10366277491481201840528644648, 7.9009971758332200260262502121, 8.550825530960030310455990640469, 9.1941073217878056387628715987, 9.612126895831645018630116035749, 10.68384706211987537813175664480, 11.36384104691768530377561933570, 11.871721088096359802539724963534, 12.695452479379286723335636830974, 13.15864480343174243363052781509, 13.65374786091395312709753190498, 14.51542539109902342453391754021, 15.08891267517869326623381045166, 15.487582372798271656175851606146, 16.72372444940571123766015892076, 17.66151466823353432551543592415

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