Properties

Label 2-21e2-1.1-c3-0-15
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 + 12·5-s − 15·8-s + 12·10-s − 20·11-s + 84·13-s + 41·16-s − 96·17-s − 12·19-s − 84·20-s − 20·22-s + 176·23-s + 19·25-s + 84·26-s − 58·29-s + 264·31-s + 161·32-s − 96·34-s + 258·37-s − 12·38-s − 180·40-s + 156·43-s + 140·44-s + 176·46-s − 408·47-s + 19·50-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s + 1.07·5-s − 0.662·8-s + 0.379·10-s − 0.548·11-s + 1.79·13-s + 0.640·16-s − 1.36·17-s − 0.144·19-s − 0.939·20-s − 0.193·22-s + 1.59·23-s + 0.151·25-s + 0.633·26-s − 0.371·29-s + 1.52·31-s + 0.889·32-s − 0.484·34-s + 1.14·37-s − 0.0512·38-s − 0.711·40-s + 0.553·43-s + 0.479·44-s + 0.564·46-s − 1.26·47-s + 0.0537·50-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

\(L(2)\) \(\approx\) \(2.259067360\)
\(L(\frac12)\) \(\approx\) \(2.259067360\)
\(L(\frac{5}{2})\) 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 \( 1 - 12 T + p^{3} T^{2} \)
11 \( 1 + 20 T + p^{3} T^{2} \)
13 \( 1 - 84 T + p^{3} T^{2} \)
17 \( 1 + 96 T + p^{3} T^{2} \)
19 \( 1 + 12 T + p^{3} T^{2} \)
23 \( 1 - 176 T + p^{3} T^{2} \)
29 \( 1 + 2 p T + p^{3} T^{2} \)
31 \( 1 - 264 T + p^{3} T^{2} \)
37 \( 1 - 258 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 - 156 T + p^{3} T^{2} \)
47 \( 1 + 408 T + p^{3} T^{2} \)
53 \( 1 - 722 T + p^{3} T^{2} \)
59 \( 1 - 492 T + p^{3} T^{2} \)
61 \( 1 - 492 T + p^{3} T^{2} \)
67 \( 1 - 412 T + p^{3} T^{2} \)
71 \( 1 + 296 T + p^{3} T^{2} \)
73 \( 1 + 240 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 - 924 T + p^{3} T^{2} \)
89 \( 1 + 744 T + p^{3} T^{2} \)
97 \( 1 - 168 T + p^{3} T^{2} \)
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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

−10.67815266956064470449255063869, −9.720097434751266129744807351014, −8.899200367275428946095094826271, −8.286616815709457462520665764157, −6.65208169021092965444863570054, −5.88328519284986765021372472819, −4.96786872420048380393928262120, −3.89164755210778366996286276561, −2.55685084019440751781184898812, −0.958533391421247021941804626295, 0.958533391421247021941804626295, 2.55685084019440751781184898812, 3.89164755210778366996286276561, 4.96786872420048380393928262120, 5.88328519284986765021372472819, 6.65208169021092965444863570054, 8.286616815709457462520665764157, 8.899200367275428946095094826271, 9.720097434751266129744807351014, 10.67815266956064470449255063869

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