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Properties

Label 2-3267-297.104-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.508 - 0.860i$
Analytic cond. $1.63044$
Root an. cond. $1.27688$
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  + (−0.961 − 0.275i)3-s + (−0.997 − 0.0697i)4-s + (−0.603 + 1.13i)5-s + (0.848 + 0.529i)9-s + (0.939 + 0.342i)12-s + (0.893 − 0.924i)15-s + (0.990 + 0.139i)16-s + (0.681 − 1.09i)20-s + (1.11 − 1.32i)23-s + (−0.364 − 0.541i)25-s + (−0.669 − 0.743i)27-s + (−0.704 + 1.74i)31-s + (−0.809 − 0.587i)36-s + (0.160 + 1.52i)37-s + (−1.11 + 0.642i)45-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)3-s + (−0.997 − 0.0697i)4-s + (−0.603 + 1.13i)5-s + (0.848 + 0.529i)9-s + (0.939 + 0.342i)12-s + (0.893 − 0.924i)15-s + (0.990 + 0.139i)16-s + (0.681 − 1.09i)20-s + (1.11 − 1.32i)23-s + (−0.364 − 0.541i)25-s + (−0.669 − 0.743i)27-s + (−0.704 + 1.74i)31-s + (−0.809 − 0.587i)36-s + (0.160 + 1.52i)37-s + (−1.11 + 0.642i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.508 - 0.860i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (995, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.508 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3989535026\)
\(L(\frac12)\) \(\approx\) \(0.3989535026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 \)
good2 \( 1 + (0.997 + 0.0697i)T^{2} \)
5 \( 1 + (0.603 - 1.13i)T + (-0.559 - 0.829i)T^{2} \)
7 \( 1 + (-0.882 - 0.469i)T^{2} \)
13 \( 1 + (-0.241 + 0.970i)T^{2} \)
17 \( 1 + (0.104 - 0.994i)T^{2} \)
19 \( 1 + (-0.978 - 0.207i)T^{2} \)
23 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.848 + 0.529i)T^{2} \)
31 \( 1 + (0.704 - 1.74i)T + (-0.719 - 0.694i)T^{2} \)
37 \( 1 + (-0.160 - 1.52i)T + (-0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.848 - 0.529i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.137 + 1.96i)T + (-0.990 + 0.139i)T^{2} \)
53 \( 1 + (-0.650 + 0.211i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.614 - 0.299i)T + (0.615 - 0.788i)T^{2} \)
61 \( 1 + (-0.719 + 0.694i)T^{2} \)
67 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (1.46 - 1.31i)T + (0.104 - 0.994i)T^{2} \)
73 \( 1 + (0.669 + 0.743i)T^{2} \)
79 \( 1 + (-0.997 - 0.0697i)T^{2} \)
83 \( 1 + (0.241 + 0.970i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.306 - 0.163i)T + (0.559 - 0.829i)T^{2} \)
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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

−8.927406315952561871730522670143, −8.385180495442933007971145950790, −7.29633118251189970589821361337, −6.95761465700692025701315038914, −6.10801797400915228449210130319, −5.19952629487998847587249146440, −4.57870220519723866228836849380, −3.68184239196444887737771970089, −2.76970330072519435964549412500, −1.18578131130032431433079430883, 0.35094893173131666264957077807, 1.41944040574097183292308647799, 3.34199869793690623848192383937, 4.29306045866062958129295332280, 4.55549471355499585879532697311, 5.54344172506659415394754150438, 5.87762602213018087825245735742, 7.34962839070075711596604510959, 7.72452324588543351406821303232, 8.854264195901192816654506350144

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