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Properties

Label 2-21e2-9.7-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.190 - 0.981i$
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  + (−0.119 + 0.207i)2-s + (1.12 + 1.31i)3-s + (0.971 + 1.68i)4-s + (−1.29 − 2.24i)5-s + (−0.407 + 0.0753i)6-s − 0.942·8-s + (−0.471 + 2.96i)9-s + 0.619·10-s + (−2.09 + 3.62i)11-s + (−1.12 + 3.17i)12-s + (1.84 + 3.18i)13-s + (1.5 − 4.23i)15-s + (−1.83 + 3.16i)16-s − 1.71·17-s + (−0.557 − 0.451i)18-s + 7.15·19-s + ⋯
L(s)  = 1  + (−0.0845 + 0.146i)2-s + (0.649 + 0.760i)3-s + (0.485 + 0.841i)4-s + (−0.579 − 1.00i)5-s + (−0.166 + 0.0307i)6-s − 0.333·8-s + (−0.157 + 0.987i)9-s + 0.195·10-s + (−0.630 + 1.09i)11-s + (−0.324 + 0.915i)12-s + (0.510 + 0.884i)13-s + (0.387 − 1.09i)15-s + (−0.457 + 0.792i)16-s − 0.414·17-s + (−0.131 − 0.106i)18-s + 1.64·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990731 + 1.20098i\)
\(L(\frac12)\) \(\approx\) \(0.990731 + 1.20098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.12 - 1.31i)T \)
7 \( 1 \)
good2 \( 1 + (0.119 - 0.207i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.29 + 2.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.84 - 3.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.71T + 17T^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + (-2.56 - 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.06 + 1.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.26 + 5.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + (5.10 + 8.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.830 + 1.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.66 + 8.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (-3.03 - 5.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 - 6.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.13 + 7.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 + 7.15T + 73T^{2} \)
79 \( 1 + (-4.91 + 8.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.44 + 5.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.03T + 89T^{2} \)
97 \( 1 + (-1.53 + 2.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.58724155651692812025966960090, −10.39115052503422742767846198161, −9.242237428220750479782724323324, −8.779726757021130604441626891789, −7.70568164166520041453444024658, −7.22315135602525132403922260306, −5.40633962881300061349178595320, −4.37932595679056281732545757237, −3.56969150965920630673385464013, −2.15546764101033458304566932672, 0.979886694820675079716434158373, 2.79057857533998739212584745006, 3.27531772151700998438125640245, 5.36840415394847064204118655454, 6.37000147189647382203061068795, 7.15918275377163859357505325698, 8.009871264722665137756864146302, 8.971263257736581726204989951494, 10.15755328369981181159977476762, 10.98399147644648253402366154159

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