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Properties

Label 2-21e2-63.4-c1-0-28
Degree $2$
Conductor $441$
Sign $-0.169 + 0.985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37146 - 1.62713i\)
\(L(\frac12)\) \(\approx\) \(1.37146 - 1.62713i\)
\(L(\frac{3}{2})\) 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

−11.16041573266512027779077693702, −10.03479940266959488818776922385, −9.661036690781083807922293437987, −8.811234606456644798036577199631, −6.50400479468265421627945579070, −5.79583286830482968443290047630, −4.76563959683769665040461769596, −3.94948027167378425628095772862, −2.76001266267839688184204914914, −1.33369048171989525893683472762, 1.85291228192384056297959678623, 3.84545812492656520123080674831, 5.13130296208713931076443849505, 5.93349882772491475329813245730, 6.51434155236879232154329866942, 7.19902525062004656661259970730, 8.348043107380856614555009895675, 9.212505378835656785435703411763, 10.50093852559701872107002927203, 11.83080069123767113628684616197

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