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Properties

Label 2-21e2-9.4-c1-0-4
Degree $2$
Conductor $441$
Sign $0.420 - 0.907i$
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  + (−0.335 − 0.580i)2-s + (−1.27 + 1.17i)3-s + (0.775 − 1.34i)4-s + (−0.712 + 1.23i)5-s + (1.10 + 0.347i)6-s − 2.38·8-s + (0.252 − 2.98i)9-s + 0.955·10-s + (2.46 + 4.27i)11-s + (0.585 + 2.62i)12-s + (−1.37 + 2.38i)13-s + (−0.537 − 2.40i)15-s + (−0.752 − 1.30i)16-s − 1.11·17-s + (−1.82 + 0.855i)18-s + 4.01·19-s + ⋯
L(s)  = 1  + (−0.236 − 0.410i)2-s + (−0.736 + 0.676i)3-s + (0.387 − 0.671i)4-s + (−0.318 + 0.551i)5-s + (0.452 + 0.141i)6-s − 0.841·8-s + (0.0843 − 0.996i)9-s + 0.302·10-s + (0.743 + 1.28i)11-s + (0.168 + 0.756i)12-s + (−0.381 + 0.661i)13-s + (−0.138 − 0.621i)15-s + (−0.188 − 0.326i)16-s − 0.271·17-s + (−0.429 + 0.201i)18-s + 0.921·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.420 - 0.907i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704443 + 0.450197i\)
\(L(\frac12)\) \(\approx\) \(0.704443 + 0.450197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.335 + 0.580i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.712 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.46 - 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 - 4.01T + 19T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-0.124 + 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.73 - 8.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.820T + 53T^{2} \)
59 \( 1 + (-3.29 + 5.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0376 + 0.0651i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.29 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (-0.922 - 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25384263345876497101177042439, −10.42949657677770087089570837040, −9.642521881795501095025208674963, −9.140964489504017571791156511829, −7.21768280173931257104952276421, −6.70749398361834659698973448593, −5.53003281845292358225984653621, −4.52471188927355595973429260649, −3.25794275020231373440652957692, −1.58042846620179711125583325010, 0.64513744929630884763687525619, 2.65571814213725303635442765674, 4.11993068766496467300844698175, 5.54878584131736739889478633847, 6.34880775568527498460485192278, 7.25485844860700310328384955007, 8.208869468216813213033470395542, 8.677587534272673140426937834793, 10.14698978728742899128608258294, 11.30631734800172976947035087264

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