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Properties

Label 2-21e2-21.17-c3-0-1
Degree $2$
Conductor $441$
Sign $-0.0285 - 0.999i$
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  + (2.44 − 1.41i)2-s + (−10.5 − 18.2i)5-s + 22.6i·8-s + (−51.6 − 29.7i)10-s + (−13.4 − 7.77i)11-s + 29.7i·13-s + (32.0 + 55.4i)16-s + (31.6 − 54.7i)17-s + (−77.4 + 44.6i)19-s − 44·22-s + (−67.3 + 38.8i)23-s + (−159.5 + 276. i)25-s + (42.1 + 72.9i)26-s + 125. i·29-s + (206. + 119. i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (−0.942 − 1.63i)5-s + 0.999i·8-s + (−1.63 − 0.942i)10-s + (−0.369 − 0.213i)11-s + 0.635i·13-s + (0.500 + 0.866i)16-s + (0.450 − 0.781i)17-s + (−0.934 + 0.539i)19-s − 0.426·22-s + (−0.610 + 0.352i)23-s + (−1.27 + 2.21i)25-s + (0.317 + 0.550i)26-s + 0.805i·29-s + (1.19 + 0.690i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0285 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0285 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0285 - 0.999i$
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: \((2,\ 441,\ (\ :3/2),\ -0.0285 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6636518734\)
\(L(\frac12)\) \(\approx\) \(0.6636518734\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.44 + 1.41i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (10.5 + 18.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (13.4 + 7.77i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 29.7iT - 2.19e3T^{2} \)
17 \( 1 + (-31.6 + 54.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (77.4 - 44.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (67.3 - 38.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 125. iT - 2.43e4T^{2} \)
31 \( 1 + (-206. - 119. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-92 - 159. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 105.T + 6.89e4T^{2} \)
43 \( 1 + 190T + 7.95e4T^{2} \)
47 \( 1 + (21.0 + 36.4i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-309. - 178. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (42.1 - 72.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (567. - 327. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (148 - 256. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 329. iT - 3.57e5T^{2} \)
73 \( 1 + (696. + 402. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (418 + 723. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + (347. + 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 566. iT - 9.12e5T^{2} \)
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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

−11.48353908273897310704317179281, −10.14696323273002726917084664517, −8.819216722327884339004180300449, −8.402284581512417364694164866828, −7.44477221663963690136241574081, −5.76901290380268888483735033290, −4.77512287687195131521665951695, −4.26950081096843390056691511106, −3.14370983439190634045519629603, −1.47969084520346029910599539815, 0.16189085231914498158178357129, 2.59602891649398609464006440685, 3.68846137330450108568002149701, 4.48676901779589407495789085984, 5.93008953019522521309814547398, 6.55049779724513568692159834825, 7.49479277677987700282231331150, 8.254626226629494999115677910036, 9.983382169277744578094851859368, 10.43993861111331237113314583407

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