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Properties

Label 2-21e2-63.38-c1-0-12
Degree $2$
Conductor $441$
Sign $0.412 - 0.910i$
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  + 1.48i·2-s + (−0.995 − 1.41i)3-s − 0.203·4-s + (−0.154 + 0.267i)5-s + (2.10 − 1.47i)6-s + 2.66i·8-s + (−1.01 + 2.82i)9-s + (−0.396 − 0.228i)10-s + (2.73 − 1.58i)11-s + (0.202 + 0.288i)12-s + (3.00 − 1.73i)13-s + (0.532 − 0.0472i)15-s − 4.36·16-s + (−2.44 + 4.22i)17-s + (−4.18 − 1.51i)18-s + (4.62 − 2.67i)19-s + ⋯
L(s)  = 1  + 1.04i·2-s + (−0.574 − 0.818i)3-s − 0.101·4-s + (−0.0689 + 0.119i)5-s + (0.859 − 0.603i)6-s + 0.942i·8-s + (−0.339 + 0.940i)9-s + (−0.125 − 0.0723i)10-s + (0.825 − 0.476i)11-s + (0.0585 + 0.0833i)12-s + (0.833 − 0.481i)13-s + (0.137 − 0.0121i)15-s − 1.09·16-s + (−0.592 + 1.02i)17-s + (−0.987 − 0.356i)18-s + (1.06 − 0.612i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12067 + 0.722758i\)
\(L(\frac12)\) \(\approx\) \(1.12067 + 0.722758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 + 1.41i)T \)
7 \( 1 \)
good2 \( 1 - 1.48iT - 2T^{2} \)
5 \( 1 + (0.154 - 0.267i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.73 + 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.00 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.44 - 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.17 - 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.70 - 1.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.52iT - 31T^{2} \)
37 \( 1 + (5.92 + 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.58 - 4.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.75 + 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 + (-0.0740 - 0.0427i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 + 5.42iT - 61T^{2} \)
67 \( 1 + 0.110T + 67T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + (8.32 + 4.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + (4.42 - 7.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.936 + 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.9 + 6.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.07454901847669290585959939319, −10.90833586731120540383387306891, −9.026531967665527684698487843823, −8.360176999389816812061058787906, −7.26258137024552951794526641285, −6.73540127849147640975603687408, −5.84537752216667607027223170037, −5.06590862268074355001296409356, −3.23880391425973042584521418629, −1.44073301472540093307255543170, 1.09448851246263100524578132200, 2.85469122647320959025022189593, 3.97975609719265264778846840668, 4.79234433793999635599061946984, 6.26013878998305469624197345966, 6.99343328667261885262864680807, 8.692435833363654913409608619926, 9.529627692129513589132231300919, 10.13781135197450792109659953042, 11.15109780635436136289351123598

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