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Properties

Label 2-264-264.35-c0-0-1
Degree $2$
Conductor $264$
Sign $-0.605 + 0.795i$
Analytic cond. $0.131753$
Root an. cond. $0.362978$
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)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 0.999·6-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (0.309 + 0.951i)18-s + (−0.690 − 0.951i)19-s + (−0.309 − 0.951i)22-s + (0.809 + 0.587i)24-s + (0.809 − 0.587i)25-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 0.999·6-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (0.309 + 0.951i)18-s + (−0.690 − 0.951i)19-s + (−0.309 − 0.951i)22-s + (0.809 + 0.587i)24-s + (0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(0.131753\)
Root analytic conductor: \(0.362978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7301745034\)
\(L(\frac12)\) \(\approx\) \(0.7301745034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
97 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.88233636631081042980346052000, −11.13075773176848812876408049547, −10.31279379892031441806253012712, −8.923048755984395808721037115802, −8.205302176560563397974406158910, −6.61022323789624614754875095602, −5.82870264069521741180257457460, −4.43237380821548307174074806690, −2.97202891575827748818381179046, −1.44443085697744619131722888894, 3.34783083246382105995178614886, 4.45910169366013068365018126965, 5.36011549215479226791635819887, 6.45624228766588268368982511414, 7.48763754118740587582700584443, 8.824792026872531034906312972503, 9.459606869538502592406388070435, 10.46196037104065699756192326924, 11.79955328584549498905148341959, 12.39877852377842348717901158249

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