Properties

Label 2-21e2-1.1-c3-0-16
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 7·4-s + 16·5-s − 15·8-s + 16·10-s + 8·11-s − 28·13-s + 41·16-s + 54·17-s + 110·19-s − 112·20-s + 8·22-s − 48·23-s + 131·25-s − 28·26-s + 110·29-s − 12·31-s + 161·32-s + 54·34-s − 246·37-s + 110·38-s − 240·40-s + 182·41-s + 128·43-s − 56·44-s − 48·46-s + 324·47-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s + 1.43·5-s − 0.662·8-s + 0.505·10-s + 0.219·11-s − 0.597·13-s + 0.640·16-s + 0.770·17-s + 1.32·19-s − 1.25·20-s + 0.0775·22-s − 0.435·23-s + 1.04·25-s − 0.211·26-s + 0.704·29-s − 0.0695·31-s + 0.889·32-s + 0.272·34-s − 1.09·37-s + 0.469·38-s − 0.948·40-s + 0.693·41-s + 0.453·43-s − 0.191·44-s − 0.153·46-s + 1.00·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.442273158\)
\(L(\frac12)\) \(\approx\) \(2.442273158\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + p^{3} T^{2} \)
5 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 - 8 T + p^{3} T^{2} \)
13 \( 1 + 28 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 - 110 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 - 110 T + p^{3} T^{2} \)
31 \( 1 + 12 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 - 182 T + p^{3} T^{2} \)
43 \( 1 - 128 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 - 162 T + p^{3} T^{2} \)
59 \( 1 - 810 T + p^{3} T^{2} \)
61 \( 1 - 8 p T + p^{3} T^{2} \)
67 \( 1 - 244 T + p^{3} T^{2} \)
71 \( 1 - 768 T + p^{3} T^{2} \)
73 \( 1 - 702 T + p^{3} T^{2} \)
79 \( 1 - 440 T + p^{3} T^{2} \)
83 \( 1 + 1302 T + p^{3} T^{2} \)
89 \( 1 - 730 T + p^{3} T^{2} \)
97 \( 1 + 294 T + p^{3} T^{2} \)
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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

−10.37464744667992854345429055478, −9.731316161941043275735257808512, −9.184730370766004090981306994543, −8.064418216046948322861954281263, −6.80409224586724947091218344125, −5.59519732042927825561725250887, −5.22546149516235726764984482624, −3.82588388650243004647961990927, −2.53378105688749247061814677342, −1.00771462962324619335121580984, 1.00771462962324619335121580984, 2.53378105688749247061814677342, 3.82588388650243004647961990927, 5.22546149516235726764984482624, 5.59519732042927825561725250887, 6.80409224586724947091218344125, 8.064418216046948322861954281263, 9.184730370766004090981306994543, 9.731316161941043275735257808512, 10.37464744667992854345429055478

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