Properties

Label 2-21e2-63.16-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.951 + 0.308i$
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  + (1.23 + 2.13i)2-s + (0.933 + 1.45i)3-s + (−2.02 + 3.51i)4-s − 2.59·5-s + (−1.96 + 3.78i)6-s − 5.05·8-s + (−1.25 + 2.72i)9-s + (−3.19 − 5.52i)10-s + 4.51·11-s + (−7.01 + 0.319i)12-s + (−0.5 − 0.866i)13-s + (−2.42 − 3.78i)15-s + (−2.16 − 3.74i)16-s + (0.472 + 0.819i)17-s + (−7.35 + 0.671i)18-s + (2.02 − 3.51i)19-s + ⋯
L(s)  = 1  + (0.869 + 1.50i)2-s + (0.538 + 0.842i)3-s + (−1.01 + 1.75i)4-s − 1.15·5-s + (−0.800 + 1.54i)6-s − 1.78·8-s + (−0.419 + 0.907i)9-s + (−1.00 − 1.74i)10-s + 1.36·11-s + (−2.02 + 0.0923i)12-s + (−0.138 − 0.240i)13-s + (−0.625 − 0.977i)15-s + (−0.540 − 0.936i)16-s + (0.114 + 0.198i)17-s + (−1.73 + 0.158i)18-s + (0.465 − 0.805i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.951 + 0.308i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312863 - 1.97980i\)
\(L(\frac12)\) \(\approx\) \(0.312863 - 1.97980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.933 - 1.45i)T \)
7 \( 1 \)
good2 \( 1 + (-1.23 - 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.472 - 0.819i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.273T + 23T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.16 - 2.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.890 - 1.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.20 - 5.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.08 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.13 - 5.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.36 + 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + (-0.753 - 1.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.472 + 0.819i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.17 + 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.78566019903927401521289175391, −10.82623489375991825098592606876, −9.350958288774225644443975528998, −8.672315541495150290813043939832, −7.75157738680166743229222664113, −7.09921596190234266557467739879, −5.90008504526266013016640516334, −4.75269203355369190188989303956, −4.05611834620167493487253016992, −3.27410587270251890043924365850, 1.00126377297477714791460656952, 2.33023948727570362732917775114, 3.70658191472231199285768690279, 4.01010783249070583872184249446, 5.62887096704238844682589655667, 6.91245599489688968551346733836, 7.914839022348836072544230176068, 9.029840676816690205627233129934, 9.838370762315078001225202379185, 11.16630028155712546979059273714

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