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Properties

Label 2-21e2-63.59-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.659 - 0.752i$
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  + (−0.254 − 0.146i)2-s + (1.27 + 1.17i)3-s + (−0.956 − 1.65i)4-s − 3.06·5-s + (−0.152 − 0.485i)6-s + 1.15i·8-s + (0.254 + 2.98i)9-s + (0.778 + 0.449i)10-s + 3.89i·11-s + (0.721 − 3.23i)12-s + (−2.02 − 1.17i)13-s + (−3.90 − 3.58i)15-s + (−1.74 + 3.02i)16-s + (−1.68 + 2.91i)17-s + (0.374 − 0.798i)18-s + (−2.20 + 1.27i)19-s + ⋯
L(s)  = 1  + (−0.179 − 0.103i)2-s + (0.736 + 0.676i)3-s + (−0.478 − 0.828i)4-s − 1.36·5-s + (−0.0622 − 0.198i)6-s + 0.406i·8-s + (0.0848 + 0.996i)9-s + (0.246 + 0.142i)10-s + 1.17i·11-s + (0.208 − 0.933i)12-s + (−0.562 − 0.324i)13-s + (−1.00 − 0.925i)15-s + (−0.436 + 0.755i)16-s + (−0.408 + 0.706i)17-s + (0.0882 − 0.188i)18-s + (−0.506 + 0.292i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.659 - 0.752i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.659 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232188 + 0.512274i\)
\(L(\frac12)\) \(\approx\) \(0.232188 + 0.512274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.27 - 1.17i)T \)
7 \( 1 \)
good2 \( 1 + (0.254 + 0.146i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.06T + 5T^{2} \)
11 \( 1 - 3.89iT - 11T^{2} \)
13 \( 1 + (2.02 + 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.98iT - 23T^{2} \)
29 \( 1 + (3.67 - 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.409 + 0.236i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.12 + 5.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.06 - 3.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.02 + 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.99 + 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.34 - 4.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 - 0.800i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.787 + 1.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (-0.856 - 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.63 + 8.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.49 - 9.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.98 + 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.13536319657340953480871580091, −10.47430594567438836928532619240, −9.639471011045323868259461839305, −8.831850482877202522826217376722, −7.955291810741656615710055084193, −7.14093511728950086872269317181, −5.44720914428318145970044551539, −4.43803504369064766088764452381, −3.78521252432986345944204235177, −2.07888479826299059469875094088, 0.33834007679381566134480424063, 2.76800404033683210404843367586, 3.67192699214280825462453134154, 4.61297487192218933799191939275, 6.51192981071616632314146013639, 7.40367408462471176186830801639, 8.072567271137815654099142357280, 8.688886234078974002485786674942, 9.473510647641838123330612248156, 11.08276314550951882800231986701

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