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Properties

Label 2-21e2-49.16-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.704 - 0.709i$
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  + (0.633 + 1.61i)2-s + (−0.738 + 0.685i)4-s + (−2.02 + 1.38i)5-s + (2.55 + 0.669i)7-s + (1.55 + 0.746i)8-s + (−3.51 − 2.39i)10-s + (1.16 + 0.175i)11-s + (−3.07 + 3.85i)13-s + (0.541 + 4.55i)14-s + (−0.373 + 4.98i)16-s + (0.121 − 0.0374i)17-s + (0.786 − 1.36i)19-s + (0.549 − 2.40i)20-s + (0.454 + 1.98i)22-s + (−4.92 − 1.51i)23-s + ⋯
L(s)  = 1  + (0.448 + 1.14i)2-s + (−0.369 + 0.342i)4-s + (−0.905 + 0.617i)5-s + (0.967 + 0.253i)7-s + (0.548 + 0.263i)8-s + (−1.11 − 0.757i)10-s + (0.350 + 0.0528i)11-s + (−0.852 + 1.06i)13-s + (0.144 + 1.21i)14-s + (−0.0934 + 1.24i)16-s + (0.0294 − 0.00907i)17-s + (0.180 − 0.312i)19-s + (0.122 − 0.538i)20-s + (0.0968 + 0.424i)22-s + (−1.02 − 0.316i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.704 - 0.709i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651874 + 1.56481i\)
\(L(\frac12)\) \(\approx\) \(0.651874 + 1.56481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 0.669i)T \)
good2 \( 1 + (-0.633 - 1.61i)T + (-1.46 + 1.36i)T^{2} \)
5 \( 1 + (2.02 - 1.38i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-1.16 - 0.175i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (3.07 - 3.85i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.121 + 0.0374i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.786 + 1.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.92 + 1.51i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (0.371 - 1.62i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.98 - 6.48i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-5.76 - 2.77i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.98 + 4.80i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.92 + 7.45i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (-8.18 + 7.59i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-5.88 - 4.01i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (9.05 + 8.40i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (2.53 + 4.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.08 + 9.15i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (3.24 - 8.26i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-6.33 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.4 - 13.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (5.91 - 0.891i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + 1.74T + 97T^{2} \)
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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

−11.50990444898087420001815728237, −10.80897415977457843134047780241, −9.491856998644127392258842642326, −8.299334753563295974196388479860, −7.55986046509969548006396020209, −6.94580725078024364840330264536, −5.89820408614071312482908801756, −4.74251382257216555665303210075, −4.01860195764983350680277431525, −2.14566987446755881053663399334, 1.00991722686753946229885722930, 2.51549332061201589436036431295, 3.91571555088888151626537160967, 4.50449435864591439741312090770, 5.63487030642525542445343418952, 7.63102791745924762614830781243, 7.74367560652458644021563444876, 9.125871595550544726159442056840, 10.25518129094574591974186394494, 10.98040656777518744743919202189

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