Properties

Label 2-21e2-147.131-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.229 - 0.973i$
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  + (0.811 − 0.318i)2-s + (−0.908 + 0.843i)4-s + (−0.228 + 0.155i)5-s + (−2.58 − 0.584i)7-s + (−1.22 + 2.54i)8-s + (−0.135 + 0.199i)10-s + (−0.640 + 4.25i)11-s + (3.73 + 2.98i)13-s + (−2.28 + 0.347i)14-s + (0.00131 − 0.0176i)16-s + (−5.10 + 1.57i)17-s + (1.61 + 0.930i)19-s + (0.0764 − 0.334i)20-s + (0.833 + 3.65i)22-s + (−0.312 + 1.01i)23-s + ⋯
L(s)  = 1  + (0.573 − 0.225i)2-s + (−0.454 + 0.421i)4-s + (−0.102 + 0.0697i)5-s + (−0.975 − 0.220i)7-s + (−0.433 + 0.899i)8-s + (−0.0429 + 0.0630i)10-s + (−0.193 + 1.28i)11-s + (1.03 + 0.826i)13-s + (−0.609 + 0.0928i)14-s + (0.000329 − 0.00440i)16-s + (−1.23 + 0.381i)17-s + (0.369 + 0.213i)19-s + (0.0170 − 0.0748i)20-s + (0.177 + 0.778i)22-s + (−0.0652 + 0.211i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658313 + 0.831931i\)
\(L(\frac12)\) \(\approx\) \(0.658313 + 0.831931i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.58 + 0.584i)T \)
good2 \( 1 + (-0.811 + 0.318i)T + (1.46 - 1.36i)T^{2} \)
5 \( 1 + (0.228 - 0.155i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (0.640 - 4.25i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (-3.73 - 2.98i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (5.10 - 1.57i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-1.61 - 0.930i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.312 - 1.01i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (3.23 + 0.737i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.59 + 2.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.88 + 5.46i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (5.58 + 2.69i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.74 + 4.69i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.78 - 9.64i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (2.18 + 2.35i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-10.0 - 6.84i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-5.45 + 5.88i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.64 - 6.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.23 - 0.965i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.31 + 1.69i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + (-1.86 + 3.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.641 - 0.804i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-15.0 + 2.26i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−11.56585621575578244912998805827, −10.58620725217341576580703585541, −9.431072773541281679246551006522, −8.889762807307951660850466997109, −7.60822867457787825230860182538, −6.70880595102392004022658499594, −5.57647835412087185224920899166, −4.25567517635804196766275160950, −3.69958819430875747291967957520, −2.22761943252480186919636129806, 0.55469036479192190586821792482, 2.99378002275762709903503769745, 3.92110493025536942643114820045, 5.22445225651178236175350945351, 6.07354188066054408685830722503, 6.72875356684772366950340111749, 8.370030136928725556564872491607, 8.941722720262190534048398989766, 10.01560469970202252335487564220, 10.79010456558698898637088171759

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