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Properties

Label 2-21e2-21.17-c3-0-13
Degree $2$
Conductor $441$
Sign $0.496 - 0.867i$
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  + (−3.21 + 1.85i)2-s + (2.87 − 4.98i)4-s + (2.27 + 3.93i)5-s − 8.32i·8-s + (−14.6 − 8.43i)10-s + (−39.4 − 22.7i)11-s + 11.6i·13-s + (38.4 + 66.6i)16-s + (−45.7 + 79.2i)17-s + (121. − 70.0i)19-s + 26.1·20-s + 168.·22-s + (−38.9 + 22.5i)23-s + (52.1 − 90.3i)25-s + (−21.6 − 37.4i)26-s + ⋯
L(s)  = 1  + (−1.13 + 0.655i)2-s + (0.359 − 0.623i)4-s + (0.203 + 0.352i)5-s − 0.367i·8-s + (−0.461 − 0.266i)10-s + (−1.08 − 0.623i)11-s + 0.249i·13-s + (0.600 + 1.04i)16-s + (−0.652 + 1.13i)17-s + (1.46 − 0.845i)19-s + 0.292·20-s + 1.63·22-s + (−0.353 + 0.204i)23-s + (0.417 − 0.722i)25-s + (−0.163 − 0.282i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.496 - 0.867i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.496 - 0.867i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8490449843\)
\(L(\frac12)\) \(\approx\) \(0.8490449843\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.21 - 1.85i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-2.27 - 3.93i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (39.4 + 22.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 11.6iT - 2.19e3T^{2} \)
17 \( 1 + (45.7 - 79.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-121. + 70.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (38.9 - 22.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 47.7iT - 2.43e4T^{2} \)
31 \( 1 + (206. + 119. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-74.4 - 128. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 393.T + 6.89e4T^{2} \)
43 \( 1 - 412.T + 7.95e4T^{2} \)
47 \( 1 + (204. + 353. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-144. - 83.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (68.7 - 119. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-280. + 161. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (212. - 367. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 727. iT - 3.57e5T^{2} \)
73 \( 1 + (-949. - 548. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-334. - 579. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 199.T + 5.71e5T^{2} \)
89 \( 1 + (-403. - 699. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 701. iT - 9.12e5T^{2} \)
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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

−10.63635660234928559142966176127, −9.781685836446038788170883084270, −8.971752175022435820311655998718, −8.091813774009025651829568957757, −7.38489589143615941510588658998, −6.40473592170829481004979550769, −5.48749660257898805372113536902, −3.91670595658817618020956972963, −2.47245835824021726315641870597, −0.72353940469175960000764583007, 0.66094879872034043540012435989, 1.96306912293414483005335308660, 3.07190645436188311855652907616, 4.86243998713346558181071848539, 5.63570735943334714409277347682, 7.38086262155616564662287237950, 7.84957477429446648206733288516, 9.196838539476120938535942638520, 9.451717428521032021733093989727, 10.52999362837245120009620938067

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