Properties

Label 2-21e2-1.1-c3-0-46
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.53·2-s + 4.46·4-s + 2.07·5-s − 12.4·8-s + 7.33·10-s − 49.1·11-s − 44.8·13-s − 79.7·16-s − 26.5·17-s + 77.7·19-s + 9.28·20-s − 173.·22-s − 55.7·23-s − 120.·25-s − 158.·26-s − 121.·29-s + 305.·31-s − 181.·32-s − 93.6·34-s + 77.1·37-s + 274.·38-s − 25.9·40-s − 248.·41-s − 147.·43-s − 219.·44-s − 196.·46-s − 269.·47-s + ⋯
L(s)  = 1  + 1.24·2-s + 0.558·4-s + 0.185·5-s − 0.551·8-s + 0.231·10-s − 1.34·11-s − 0.956·13-s − 1.24·16-s − 0.378·17-s + 0.938·19-s + 0.103·20-s − 1.68·22-s − 0.505·23-s − 0.965·25-s − 1.19·26-s − 0.777·29-s + 1.77·31-s − 1.00·32-s − 0.472·34-s + 0.342·37-s + 1.17·38-s − 0.102·40-s − 0.947·41-s − 0.521·43-s − 0.753·44-s − 0.630·46-s − 0.837·47-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: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.53T + 8T^{2} \)
5 \( 1 - 2.07T + 125T^{2} \)
11 \( 1 + 49.1T + 1.33e3T^{2} \)
13 \( 1 + 44.8T + 2.19e3T^{2} \)
17 \( 1 + 26.5T + 4.91e3T^{2} \)
19 \( 1 - 77.7T + 6.85e3T^{2} \)
23 \( 1 + 55.7T + 1.21e4T^{2} \)
29 \( 1 + 121.T + 2.43e4T^{2} \)
31 \( 1 - 305.T + 2.97e4T^{2} \)
37 \( 1 - 77.1T + 5.06e4T^{2} \)
41 \( 1 + 248.T + 6.89e4T^{2} \)
43 \( 1 + 147.T + 7.95e4T^{2} \)
47 \( 1 + 269.T + 1.03e5T^{2} \)
53 \( 1 - 141.T + 1.48e5T^{2} \)
59 \( 1 - 424.T + 2.05e5T^{2} \)
61 \( 1 + 587.T + 2.26e5T^{2} \)
67 \( 1 + 179.T + 3.00e5T^{2} \)
71 \( 1 + 674.T + 3.57e5T^{2} \)
73 \( 1 - 237.T + 3.89e5T^{2} \)
79 \( 1 - 495.T + 4.93e5T^{2} \)
83 \( 1 - 24.4T + 5.71e5T^{2} \)
89 \( 1 + 1.07e3T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−10.24857434040480258653431088824, −9.567800984833608257204516204756, −8.270914522848226906583348084408, −7.32208397941628594517430170046, −6.13394558074209423450318710344, −5.26300228119048885105944645669, −4.55539214101935584050561444063, −3.24468819522281607746867492661, −2.27823610644499134484477891011, 0, 2.27823610644499134484477891011, 3.24468819522281607746867492661, 4.55539214101935584050561444063, 5.26300228119048885105944645669, 6.13394558074209423450318710344, 7.32208397941628594517430170046, 8.270914522848226906583348084408, 9.567800984833608257204516204756, 10.24857434040480258653431088824

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