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Properties

Label 2-3267-297.178-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.780 + 0.625i$
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  + (−0.241 + 0.970i)3-s + (0.438 − 0.898i)4-s + (0.0121 − 0.347i)5-s + (−0.882 − 0.469i)9-s + (0.766 + 0.642i)12-s + (0.333 + 0.0957i)15-s + (−0.615 − 0.788i)16-s + (−0.306 − 0.163i)20-s + (−0.173 − 0.984i)23-s + (0.877 + 0.0613i)25-s + (0.669 − 0.743i)27-s + (1.51 + 0.213i)31-s + (−0.809 + 0.587i)36-s + (−0.0363 + 0.345i)37-s + (−0.173 + 0.300i)45-s + ⋯
L(s)  = 1  + (−0.241 + 0.970i)3-s + (0.438 − 0.898i)4-s + (0.0121 − 0.347i)5-s + (−0.882 − 0.469i)9-s + (0.766 + 0.642i)12-s + (0.333 + 0.0957i)15-s + (−0.615 − 0.788i)16-s + (−0.306 − 0.163i)20-s + (−0.173 − 0.984i)23-s + (0.877 + 0.0613i)25-s + (0.669 − 0.743i)27-s + (1.51 + 0.213i)31-s + (−0.809 + 0.587i)36-s + (−0.0363 + 0.345i)37-s + (−0.173 + 0.300i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193839680\)
\(L(\frac12)\) \(\approx\) \(1.193839680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.241 - 0.970i)T \)
11 \( 1 \)
good2 \( 1 + (-0.438 + 0.898i)T^{2} \)
5 \( 1 + (-0.0121 + 0.347i)T + (-0.997 - 0.0697i)T^{2} \)
7 \( 1 + (-0.0348 + 0.999i)T^{2} \)
13 \( 1 + (0.719 - 0.694i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.882 - 0.469i)T^{2} \)
31 \( 1 + (-1.51 - 0.213i)T + (0.961 + 0.275i)T^{2} \)
37 \( 1 + (0.0363 - 0.345i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (0.882 + 0.469i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.823 + 1.68i)T + (-0.615 + 0.788i)T^{2} \)
53 \( 1 + (-0.473 + 1.45i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.856 - 1.27i)T + (-0.374 + 0.927i)T^{2} \)
61 \( 1 + (-0.961 + 0.275i)T^{2} \)
67 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (1.25 - 1.39i)T + (-0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.669 + 0.743i)T^{2} \)
79 \( 1 + (-0.438 + 0.898i)T^{2} \)
83 \( 1 + (0.719 + 0.694i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.0655 + 1.87i)T + (-0.997 + 0.0697i)T^{2} \)
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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

−8.716246524798942428599973717457, −8.364952285495972095738029676060, −6.95083151738805057781106737742, −6.45700761540974467023099828023, −5.55000903132580919162518495244, −4.98011296455067868736914320845, −4.30232286945353706603505762569, −3.18437308268192175220114498579, −2.21368872522134943005625089122, −0.78825676687835719732502455731, 1.34658994825243286605514736893, 2.51828484744841125607475933042, 3.06013389209805316413962939883, 4.15799612313115770770327516847, 5.21051177076192962289869374404, 6.25997860974625992153770037987, 6.63091163695120118117300166856, 7.54792042585362251570281375794, 7.87392905804059688460807892466, 8.677439744615314809798231847705

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