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Properties

Label 2-21e2-7.2-c3-0-38
Degree $2$
Conductor $441$
Sign $0.198 + 0.980i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.20 − 2.09i)2-s + (1.08 + 1.88i)4-s + (9.94 − 17.2i)5-s + 24.5·8-s + (−24.0 − 41.6i)10-s + (11.9 + 20.7i)11-s + 87.3·13-s + (20.9 − 36.2i)16-s + (−2.81 − 4.88i)17-s + (−32.4 + 56.1i)19-s + 43.2·20-s + 57.7·22-s + (−12.7 + 22.1i)23-s + (−135. − 234. i)25-s + (105. − 182. i)26-s + ⋯
L(s)  = 1  + (0.426 − 0.739i)2-s + (0.135 + 0.235i)4-s + (0.889 − 1.54i)5-s + 1.08·8-s + (−0.759 − 1.31i)10-s + (0.328 + 0.568i)11-s + 1.86·13-s + (0.327 − 0.567i)16-s + (−0.0402 − 0.0696i)17-s + (−0.391 + 0.678i)19-s + 0.483·20-s + 0.560·22-s + (−0.116 + 0.200i)23-s + (−1.08 − 1.87i)25-s + (0.795 − 1.37i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.585708532\)
\(L(\frac12)\) \(\approx\) \(3.585708532\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.20 + 2.09i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-9.94 + 17.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-11.9 - 20.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 87.3T + 2.19e3T^{2} \)
17 \( 1 + (2.81 + 4.88i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (32.4 - 56.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (12.7 - 22.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 60.3T + 2.43e4T^{2} \)
31 \( 1 + (-61.3 - 106. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-28.0 + 48.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 299.T + 6.89e4T^{2} \)
43 \( 1 + 501.T + 7.95e4T^{2} \)
47 \( 1 + (-152. + 264. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (187. + 324. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (313. + 543. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (1.87 - 3.25i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-406. - 704. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 165.T + 3.57e5T^{2} \)
73 \( 1 + (309. + 536. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-69.1 + 119. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 621.T + 5.71e5T^{2} \)
89 \( 1 + (-142. + 247. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 603.T + 9.12e5T^{2} \)
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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

−10.59546117543750982662143727690, −9.687077053964280650060583875035, −8.723601316400144201006039113142, −8.064421789673339483105407786910, −6.56153465203783585694929606169, −5.53003537198711305594591695808, −4.49134761254252384294577137581, −3.60974801325398666362746762784, −1.93090394129773670482790925844, −1.22430955073623662555331717348, 1.47065383913897400930904364619, 2.83668587492723389470072500433, 4.07810524328721266507511360819, 5.66767602592171733384019376857, 6.26380608404244140987771138746, 6.74472369227875786893920202009, 7.889711843085458705222677068984, 9.111610619841004003779106298653, 10.20931459696996584727888178814, 10.95062156219612426823852140133

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