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Properties

Label 2-21e2-147.131-c1-0-8
Degree $2$
Conductor $441$
Sign $0.994 + 0.107i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.994 + 0.107i$
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: \((2,\ 441,\ (\ :1/2),\ 0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868198 - 0.0468744i\)
\(L(\frac12)\) \(\approx\) \(0.868198 - 0.0468744i\)
\(L(\frac{3}{2})\) 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

−10.98219075929003445428115648680, −9.943626441680640906544462483993, −9.217753878757298983625578971447, −8.529753418129927530529148365241, −7.53799795256729566962859136435, −6.54666230713029842391459552170, −5.58973264812282173132921054867, −3.94095866103809508792347013594, −3.26479514125089488407130257666, −0.868979418544816973011671512625, 1.18467220126356078468143564374, 2.86890127153711277467523019838, 4.27175242162663013624359366580, 5.53180375957216189832039643559, 6.36550230805970515800851718314, 7.67219915712716445344812076840, 8.563479747477411679496765387291, 9.590670012914396068280698976795, 10.04422836175059224210413620469, 10.78008779532617352696913766378

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