Properties

Label 2-21e2-9.4-c1-0-26
Degree $2$
Conductor $441$
Sign $-0.975 + 0.218i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 2.13i)2-s + (1.66 − 0.486i)3-s + (−2.02 + 3.51i)4-s + (1.82 − 3.16i)5-s + (−3.08 − 2.94i)6-s + 5.05·8-s + (2.52 − 1.61i)9-s − 9.00·10-s + (−0.203 − 0.351i)11-s + (−1.66 + 6.82i)12-s + (0.243 − 0.421i)13-s + (1.5 − 6.15i)15-s + (−2.16 − 3.74i)16-s + 4.85·17-s + (−6.55 − 3.39i)18-s − 1.97·19-s + ⋯
L(s)  = 1  + (−0.869 − 1.50i)2-s + (0.959 − 0.280i)3-s + (−1.01 + 1.75i)4-s + (0.817 − 1.41i)5-s + (−1.25 − 1.20i)6-s + 1.78·8-s + (0.842 − 0.538i)9-s − 2.84·10-s + (−0.0612 − 0.106i)11-s + (−0.479 + 1.96i)12-s + (0.0675 − 0.116i)13-s + (0.387 − 1.58i)15-s + (−0.540 − 0.936i)16-s + 1.17·17-s + (−1.54 − 0.800i)18-s − 0.452·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.975 + 0.218i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146167 - 1.32278i\)
\(L(\frac12)\) \(\approx\) \(0.146167 - 1.32278i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.486i)T \)
7 \( 1 \)
good2 \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.82 + 3.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.203 + 0.351i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.243 + 0.421i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.82 + 6.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.51 - 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.15 - 5.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.56T + 53T^{2} \)
59 \( 1 + (3.05 - 5.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 - 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.80 - 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 - 1.97T + 73T^{2} \)
79 \( 1 + (4.08 + 7.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.08 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (-4.74 - 8.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.37268554307342835518629972763, −9.648624544531917070843245052712, −9.176457891973046466559946984710, −8.352070603338224733065605731237, −7.69447996070222453144090012105, −5.82571208766638658243708453983, −4.39012439886547428136351327483, −3.24478376882061664185874062099, −1.96862064208432422849519764303, −1.12132291278257295863602460905, 2.11867208158155849644416967126, 3.56643457779370280742787129415, 5.27171863547060069263942395449, 6.27037429663027444782690439290, 7.09895354992448153078768793836, 7.75796207369766280477051293721, 8.735611689559725822979403368636, 9.618580395739600967230621537657, 10.16293494623733257306402689781, 10.90373897184092745125761341686

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