Properties

Label 2-21e2-1.1-c3-0-17
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  + 3·2-s + 4-s − 3·5-s − 21·8-s − 9·10-s + 15·11-s + 64·13-s − 71·16-s + 84·17-s + 16·19-s − 3·20-s + 45·22-s + 84·23-s − 116·25-s + 192·26-s + 297·29-s + 253·31-s − 45·32-s + 252·34-s − 316·37-s + 48·38-s + 63·40-s + 360·41-s + 26·43-s + 15·44-s + 252·46-s − 30·47-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.268·5-s − 0.928·8-s − 0.284·10-s + 0.411·11-s + 1.36·13-s − 1.10·16-s + 1.19·17-s + 0.193·19-s − 0.0335·20-s + 0.436·22-s + 0.761·23-s − 0.927·25-s + 1.44·26-s + 1.90·29-s + 1.46·31-s − 0.248·32-s + 1.27·34-s − 1.40·37-s + 0.204·38-s + 0.249·40-s + 1.37·41-s + 0.0922·43-s + 0.0513·44-s + 0.807·46-s − 0.0931·47-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

\(L(2)\) \(\approx\) \(3.072578329\)
\(L(\frac12)\) \(\approx\) \(3.072578329\)
\(L(\frac{5}{2})\) 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 \( 1 - 3 T + p^{3} T^{2} \)
5 \( 1 + 3 T + p^{3} T^{2} \)
11 \( 1 - 15 T + p^{3} T^{2} \)
13 \( 1 - 64 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 - 297 T + p^{3} T^{2} \)
31 \( 1 - 253 T + p^{3} T^{2} \)
37 \( 1 + 316 T + p^{3} T^{2} \)
41 \( 1 - 360 T + p^{3} T^{2} \)
43 \( 1 - 26 T + p^{3} T^{2} \)
47 \( 1 + 30 T + p^{3} T^{2} \)
53 \( 1 + 363 T + p^{3} T^{2} \)
59 \( 1 + 15 T + p^{3} T^{2} \)
61 \( 1 - 118 T + p^{3} T^{2} \)
67 \( 1 + 370 T + p^{3} T^{2} \)
71 \( 1 - 342 T + p^{3} T^{2} \)
73 \( 1 + 362 T + p^{3} T^{2} \)
79 \( 1 - 467 T + p^{3} T^{2} \)
83 \( 1 - 477 T + p^{3} T^{2} \)
89 \( 1 - 906 T + p^{3} T^{2} \)
97 \( 1 + 503 T + p^{3} T^{2} \)
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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

−10.92741559891106202143646147673, −9.840245874034159708521203948657, −8.817469973780390783242943074661, −8.003616517465058048823210060680, −6.61482869831534676779707879266, −5.87080526465409933256741657776, −4.81014610877187189424899088866, −3.82275054002836987472587538824, −2.98851310927051577184490502479, −1.02176902159605660452794947887, 1.02176902159605660452794947887, 2.98851310927051577184490502479, 3.82275054002836987472587538824, 4.81014610877187189424899088866, 5.87080526465409933256741657776, 6.61482869831534676779707879266, 8.003616517465058048823210060680, 8.817469973780390783242943074661, 9.840245874034159708521203948657, 10.92741559891106202143646147673

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