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Properties

Label 2-21e2-63.25-c1-0-32
Degree $2$
Conductor $441$
Sign $0.605 + 0.795i$
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  + 2.53·2-s + (−0.592 − 1.62i)3-s + 4.41·4-s + (0.439 − 0.761i)5-s + (−1.50 − 4.12i)6-s + 6.10·8-s + (−2.29 + 1.92i)9-s + (1.11 − 1.92i)10-s + (−1.93 − 3.35i)11-s + (−2.61 − 7.18i)12-s + (2.72 + 4.72i)13-s + (−1.50 − 0.264i)15-s + 6.63·16-s + (−0.826 + 1.43i)17-s + (−5.81 + 4.88i)18-s + (−1.20 − 2.08i)19-s + ⋯
L(s)  = 1  + 1.79·2-s + (−0.342 − 0.939i)3-s + 2.20·4-s + (0.196 − 0.340i)5-s + (−0.612 − 1.68i)6-s + 2.15·8-s + (−0.766 + 0.642i)9-s + (0.352 − 0.609i)10-s + (−0.584 − 1.01i)11-s + (−0.754 − 2.07i)12-s + (0.756 + 1.30i)13-s + (−0.387 − 0.0682i)15-s + 1.65·16-s + (−0.200 + 0.347i)17-s + (−1.37 + 1.15i)18-s + (−0.276 − 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06473 - 1.51927i\)
\(L(\frac12)\) \(\approx\) \(3.06473 - 1.51927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.592 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 - 2.53T + 2T^{2} \)
5 \( 1 + (-0.439 + 0.761i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.93 + 3.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.826 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.58 - 2.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.02 + 5.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 + (-2.27 - 3.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.592 - 1.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0923 - 0.160i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.02T + 47T^{2} \)
53 \( 1 + (3.64 - 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.66T + 59T^{2} \)
61 \( 1 + 2.59T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + (-6.39 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.95T + 79T^{2} \)
83 \( 1 + (-0.109 + 0.189i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.51 + 9.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.25 - 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36179222844986759844652931147, −10.79606787799903534344595552340, −9.004387640974797185214027976300, −7.903684620168837491660079716635, −6.76706679643782219469694589307, −6.10468298558753069554182589125, −5.34966852562544007477651273867, −4.24520980253784602059303601803, −2.95724197247819951691978530924, −1.68847135581890028304984931045, 2.52109541255825529462245258740, 3.51186819505681221662884877115, 4.52269838339875983186760250700, 5.34202822079052198010951961552, 6.10172851736426718026058978671, 7.09331127363754760564468015069, 8.432206991355089071189491768463, 9.975889026443323201410220872855, 10.62746871174361823345203466699, 11.26037504950219617903092292694

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