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Properties

Label 2-21e2-7.4-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.266 - 0.963i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 441,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36065 + 1.78855i\)
\(L(\frac12)\) \(\approx\) \(1.36065 + 1.78855i\)
\(L(\frac{3}{2})\) 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

−11.02090591274987117362807617047, −10.71931607229873905240695053482, −9.608106847756638524911174073172, −8.435306804017051460939841793516, −7.30784016852034462052284285322, −6.45490217576550286973194696935, −6.11017428410712228794791537419, −4.89009565842659254257570699128, −3.57164751626833891532980491553, −2.17410195544381373307158306863, 1.42585812511258605735415408409, 2.39308240494693348068871152938, 4.05027739523580402681731346377, 4.76788602342978370105735376727, 5.70767997243788889678556460443, 7.15493606615762902651882748294, 8.276255283095500725974311184833, 9.577924777195509527047162479013, 9.762108333841028322958995533774, 11.07597797889966436151275821958

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