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Properties

Label 2-3225-3225.221-c0-0-1
Degree $2$
Conductor $3225$
Sign $0.764 + 0.644i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (0.951 + 0.309i)2-s + (0.913 − 0.406i)3-s + (−0.994 + 0.104i)5-s + (0.994 − 0.104i)6-s + (0.5 + 0.866i)7-s + (−0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.978 − 0.207i)10-s + (−0.951 − 0.309i)11-s + (1.08 − 1.20i)13-s + (0.207 + 0.978i)14-s + (−0.866 + 0.499i)15-s + (−0.309 − 0.951i)16-s + (0.866 − 0.499i)18-s + (0.809 + 0.587i)21-s + (−0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.913 − 0.406i)3-s + (−0.994 + 0.104i)5-s + (0.994 − 0.104i)6-s + (0.5 + 0.866i)7-s + (−0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.978 − 0.207i)10-s + (−0.951 − 0.309i)11-s + (1.08 − 1.20i)13-s + (0.207 + 0.978i)14-s + (−0.866 + 0.499i)15-s + (−0.309 − 0.951i)16-s + (0.866 − 0.499i)18-s + (0.809 + 0.587i)21-s + (−0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.219770898\)
\(L(\frac12)\) \(\approx\) \(2.219770898\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.08 + 1.20i)T + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T^{2} \)
19 \( 1 + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (-1.20 + 1.08i)T + (0.104 - 0.994i)T^{2} \)
29 \( 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
37 \( 1 + (1.58 + 0.336i)T + (0.913 + 0.406i)T^{2} \)
41 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.669 + 0.743i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.913 - 0.406i)T^{2} \)
67 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
71 \( 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.438151571780960306223214982599, −8.246195731456126486037187778785, −7.19468773563667493795889481590, −6.53645576103776077510564617937, −5.53339827188612242470753419661, −4.97022253400119329909601628229, −4.01598802128105583408154588196, −3.15925941357447117814709271839, −2.72629385958887044498369540934, −0.982196900781724501878385516308, 1.60353772309625348787759539494, 2.83596679444557391602858556844, 3.59047962494227807834391426290, 4.19246801518430913678838943508, 4.67180639700483145325484451969, 5.46902015354470904312271366167, 6.90163246354896376221220644159, 7.50411707168026716573691144412, 8.270231384981291383188101839377, 8.725918242316300869784621593360

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