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Properties

Label 2-21e2-63.34-c2-0-33
Degree $2$
Conductor $441$
Sign $0.433 - 0.901i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.662 − 1.14i)2-s + (2.99 − 0.0895i)3-s + (1.12 + 1.94i)4-s + (−6.26 + 3.61i)5-s + (1.88 − 3.49i)6-s + 8.27·8-s + (8.98 − 0.537i)9-s + 9.58i·10-s + (−8.56 + 14.8i)11-s + (3.54 + 5.72i)12-s + (−9.05 + 5.22i)13-s + (−18.4 + 11.4i)15-s + (0.990 − 1.71i)16-s + 6.23i·17-s + (5.33 − 10.6i)18-s − 11.8i·19-s + ⋯
L(s)  = 1  + (0.331 − 0.573i)2-s + (0.999 − 0.0298i)3-s + (0.280 + 0.486i)4-s + (−1.25 + 0.723i)5-s + (0.313 − 0.583i)6-s + 1.03·8-s + (0.998 − 0.0596i)9-s + 0.958i·10-s + (−0.778 + 1.34i)11-s + (0.295 + 0.477i)12-s + (−0.696 + 0.402i)13-s + (−1.23 + 0.760i)15-s + (0.0618 − 0.107i)16-s + 0.366i·17-s + (0.296 − 0.592i)18-s − 0.625i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.433 - 0.901i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.433 - 0.901i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95866 + 1.23142i\)
\(L(\frac12)\) \(\approx\) \(1.95866 + 1.23142i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.99 + 0.0895i)T \)
7 \( 1 \)
good2 \( 1 + (-0.662 + 1.14i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (6.26 - 3.61i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (8.56 - 14.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (9.05 - 5.22i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 6.23iT - 289T^{2} \)
19 \( 1 + 11.8iT - 361T^{2} \)
23 \( 1 + (-5.85 - 10.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (5.48 - 9.50i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (24.3 - 14.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 67.4T + 1.36e3T^{2} \)
41 \( 1 + (14.1 - 8.17i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-31.2 + 54.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-26.1 - 15.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 22.6T + 2.80e3T^{2} \)
59 \( 1 + (1.58 - 0.916i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.9 - 7.49i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.7 + 51.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 6.84T + 5.04e3T^{2} \)
73 \( 1 + 87.0iT - 5.32e3T^{2} \)
79 \( 1 + (34.3 - 59.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-30.2 - 17.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 14.9iT - 7.92e3T^{2} \)
97 \( 1 + (41.2 + 23.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.10372739623439412410281944486, −10.37487172209498045780532494167, −9.315004073046999332684073567327, −8.070928446054567240043161822255, −7.34539976851586879585465373791, −7.07601540296955579402458998539, −4.70551332037357589536713093145, −3.95120197889541891553461071712, −2.95815510627998880517416855513, −2.09930614407795447255560472607, 0.77054436202753780245554438931, 2.64953232420724279337392410267, 3.94364103226046977023889756614, 4.88127127444267681743288647825, 5.94552635635639242485588857243, 7.41036543342754087760178324424, 7.84746597697643610819295128658, 8.628210191308143367453459739746, 9.752303484067059652645650073945, 10.75766098765106905929596038691

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