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Properties

Label 2-21e2-7.4-c1-0-11
Degree $2$
Conductor $441$
Sign $0.991 - 0.126i$
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  + (0.866 + 1.5i)2-s + (−0.5 + 0.866i)4-s + (−1.73 − 3i)5-s + 1.73·8-s + (3 − 5.19i)10-s + (1.73 − 3i)11-s + 2·13-s + (2.49 + 4.33i)16-s + (1.73 − 3i)17-s + (2 + 3.46i)19-s + 3.46·20-s + 6·22-s + (−1.73 − 3i)23-s + (−3.5 + 6.06i)25-s + (1.73 + 3i)26-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.250 + 0.433i)4-s + (−0.774 − 1.34i)5-s + 0.612·8-s + (0.948 − 1.64i)10-s + (0.522 − 0.904i)11-s + 0.554·13-s + (0.624 + 1.08i)16-s + (0.420 − 0.727i)17-s + (0.458 + 0.794i)19-s + 0.774·20-s + 1.27·22-s + (−0.361 − 0.625i)23-s + (−0.700 + 1.21i)25-s + (0.339 + 0.588i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85347 + 0.117621i\)
\(L(\frac12)\) \(\approx\) \(1.85347 + 0.117621i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.73 + 3i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 + 3i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-3.46 - 6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 - 6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.73 + 3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.43448843612790910054398687001, −10.19038276546425465937669822121, −8.921420986456272782243836595196, −8.246263152268771083765315648130, −7.48511243474022046708262547843, −6.26517661928882596376424624397, −5.45822745168916558583531875188, −4.52610011567401792800797087665, −3.62766619474475797541634169532, −1.12507189690769241095598355926, 1.84621836664143887549353464714, 3.22891081079250314690446454487, 3.76179095000196081587617395001, 4.96529501032693138863062532402, 6.57876256347871427496705803701, 7.28216673070781945411917891372, 8.275119335218803328969275669701, 9.807707486757128340068809268807, 10.45898096671117912247175036227, 11.32828646481232179747565664538

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