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Properties

Label 2-21e2-49.4-c1-0-15
Degree $2$
Conductor $441$
Sign $0.217 - 0.975i$
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  + (2.09 + 1.42i)2-s + (1.62 + 4.13i)4-s + (2.38 − 0.737i)5-s + (−1.40 − 2.24i)7-s + (−1.38 + 6.04i)8-s + (6.06 + 1.87i)10-s + (−0.229 + 3.05i)11-s + (−0.101 + 0.0487i)13-s + (0.270 − 6.71i)14-s + (−5.02 + 4.66i)16-s + (0.565 − 0.0852i)17-s + (−1.46 − 2.52i)19-s + (6.92 + 8.68i)20-s + (−4.85 + 6.08i)22-s + (−6.98 − 1.05i)23-s + ⋯
L(s)  = 1  + (1.48 + 1.01i)2-s + (0.811 + 2.06i)4-s + (1.06 − 0.329i)5-s + (−0.529 − 0.848i)7-s + (−0.487 + 2.13i)8-s + (1.91 + 0.591i)10-s + (−0.0690 + 0.921i)11-s + (−0.0280 + 0.0135i)13-s + (0.0724 − 1.79i)14-s + (−1.25 + 1.16i)16-s + (0.137 − 0.0206i)17-s + (−0.335 − 0.580i)19-s + (1.54 + 1.94i)20-s + (−1.03 + 1.29i)22-s + (−1.45 − 0.219i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.217 - 0.975i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.217 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56946 + 2.05915i\)
\(L(\frac12)\) \(\approx\) \(2.56946 + 2.05915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 + 2.24i)T \)
good2 \( 1 + (-2.09 - 1.42i)T + (0.730 + 1.86i)T^{2} \)
5 \( 1 + (-2.38 + 0.737i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.229 - 3.05i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (0.101 - 0.0487i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.565 + 0.0852i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (1.46 + 2.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.98 + 1.05i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (0.419 + 0.526i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-2.54 + 4.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.977 - 2.49i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (0.963 - 4.22i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.61 + 11.4i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.74 + 1.87i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (-0.511 - 1.30i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-12.2 - 3.77i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (0.199 - 0.507i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (2.29 - 3.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.460 - 0.578i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-11.2 + 7.64i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-2.00 - 3.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.54 + 3.63i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.756 - 10.0i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−11.76546288899914713117774608766, −10.30201435855375390393547929168, −9.608071819739806610082510754603, −8.219114485023239845758408511460, −7.19159053801988348305112973860, −6.48437981163515618896445356699, −5.64003947245410602888562078717, −4.65983620101381043855039716837, −3.78684140369069343100727695680, −2.28302824118427339669628505075, 1.84513275829313349906111865107, 2.79976893884653687502288811966, 3.76487174365044633611887915615, 5.21810739859387559631053934806, 5.95348983130172395362844656847, 6.44869458875214318011689148369, 8.359614307100842754825203309615, 9.666660712526066267349006545167, 10.18740478324201253557909814871, 11.13570540317248908093938950178

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