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Properties

Label 1-264-264.107-r1-0-0
Degree $1$
Conductor $264$
Sign $0.530 + 0.847i$
Analytic cond. $28.3707$
Root an. cond. $28.3707$
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  + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (−0.809 + 0.587i)25-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + (0.809 + 0.587i)37-s + (−0.809 + 0.587i)41-s − 43-s + (−0.809 + 0.587i)47-s + (0.309 + 0.951i)49-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (−0.809 + 0.587i)25-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + (0.809 + 0.587i)37-s + (−0.809 + 0.587i)41-s − 43-s + (−0.809 + 0.587i)47-s + (0.309 + 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(28.3707\)
Root analytic conductor: \(28.3707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 264,\ (1:\ ),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540755860 + 0.8537733332i\)
\(L(\frac12)\) \(\approx\) \(1.540755860 + 0.8537733332i\)
\(L(1)\) \(\approx\) \(1.092361551 + 0.1958657819i\)
\(L(1)\) \(\approx\) \(1.092361551 + 0.1958657819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−25.17337959792664609023272401108, −24.84174818988525659597248042275, −23.63211481705869093130462593068, −22.77075782187420436503131425229, −21.70072619212870633456549551413, −20.9209671496187665321594480176, −20.038422691907122328878422357354, −18.96308364386970724968975787505, −18.21425285876024571870360520938, −16.84486045110357988633555569551, −16.31722863945521940403884079471, −15.39767994234695600200172194304, −14.02139715912218390064972328774, −13.23029253018191759947550182207, −12.24368549154826299022662124399, −11.47493873133157921448390514736, −9.79732295299710158202357662170, −9.30700810484573449332926498147, −8.26066266542916349666669495714, −6.87793197813427120271617567916, −5.786675797128892993404927399131, −4.83594530357704387532202662100, −3.47895083803210121706563959635, −2.09942230432361039902249775792, −0.6549911648100846326583507446, 1.09964655373959141251811132583, 2.89237326719512227131642487755, 3.53749849352347772035601873760, 5.20741682276156566762791502126, 6.41685414436385326108457819412, 7.12478767139540180182056250173, 8.36678242963936777376407393057, 9.78774974318905550590638789412, 10.41440011999991167442271937159, 11.35001192550195465574146305172, 12.79578468524988407715168623096, 13.48206750890998453360928106792, 14.57211283901659653741751430367, 15.433740769545459378796359133773, 16.489626108359900781610992103914, 17.54628414178923510522843837917, 18.32505409058865883959115312761, 19.39635843682016871143858618164, 20.077707693938791141554673993888, 21.34311194953089258318072998502, 22.20145126448556158591744363894, 22.97053557482237617944850665154, 23.70513263622535927524328219685, 25.16990093721756612537647367992, 25.70057801463156666451588129774

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