Properties

Label 2-21e2-49.2-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.578 - 0.815i$
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  + (0.742 + 0.229i)2-s + (−1.15 − 0.786i)4-s + (−1.96 + 0.295i)5-s + (1.38 + 2.25i)7-s + (−1.64 − 2.06i)8-s + (−1.52 − 0.229i)10-s + (−4.54 + 4.21i)11-s + (1.11 + 4.87i)13-s + (0.508 + 1.99i)14-s + (0.269 + 0.687i)16-s + (−0.0440 − 0.588i)17-s + (0.327 − 0.566i)19-s + (2.49 + 1.20i)20-s + (−4.34 + 2.09i)22-s + (−0.115 + 1.53i)23-s + ⋯
L(s)  = 1  + (0.525 + 0.162i)2-s + (−0.576 − 0.393i)4-s + (−0.876 + 0.132i)5-s + (0.521 + 0.853i)7-s + (−0.581 − 0.729i)8-s + (−0.481 − 0.0726i)10-s + (−1.37 + 1.27i)11-s + (0.308 + 1.35i)13-s + (0.135 + 0.532i)14-s + (0.0674 + 0.171i)16-s + (−0.0106 − 0.142i)17-s + (0.0750 − 0.129i)19-s + (0.557 + 0.268i)20-s + (−0.925 + 0.445i)22-s + (−0.0240 + 0.320i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.578 - 0.815i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364535 + 0.705381i\)
\(L(\frac12)\) \(\approx\) \(0.364535 + 0.705381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 2.25i)T \)
good2 \( 1 + (-0.742 - 0.229i)T + (1.65 + 1.12i)T^{2} \)
5 \( 1 + (1.96 - 0.295i)T + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (4.54 - 4.21i)T + (0.822 - 10.9i)T^{2} \)
13 \( 1 + (-1.11 - 4.87i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.0440 + 0.588i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (-0.327 + 0.566i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.115 - 1.53i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-4.13 - 1.98i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (4.14 + 7.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.95 - 4.74i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (4.92 + 6.17i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-2.05 + 2.57i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-1.79 - 0.553i)T + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (-5.45 - 3.71i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (4.21 + 0.634i)T + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (-11.9 + 8.11i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (-1.01 - 1.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.23 + 3.00i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (6.44 - 1.98i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (-0.105 + 0.183i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.38 - 6.08i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-8.47 - 7.86i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−11.67821047206231422509440520621, −10.56678148168717276715482437665, −9.568987799286102003775478754857, −8.735048273115712432431358982497, −7.73371163954893263837806938361, −6.76965758164900656453332634057, −5.44057220608074925719153446160, −4.76078664943196163697159974583, −3.79500607392356903545520362573, −2.13179013784624067450082374304, 0.42129067179371713847783414905, 3.03223118350530926158895007219, 3.77726711127395089769679669035, 4.91737040755138240035357986714, 5.68769953591076829643444156690, 7.39156803197526202341261932679, 8.272566352139939297406251246737, 8.460559462132110348061394084591, 10.25651325147214836503497657341, 10.85831098340851793189433425791

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