Properties

Label 2-21e2-63.25-c1-0-1
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  + 1.34·2-s + (−1.11 + 1.32i)3-s − 0.184·4-s + (−1.26 + 2.19i)5-s + (−1.5 + 1.78i)6-s − 2.94·8-s + (−0.520 − 2.95i)9-s + (−1.70 + 2.95i)10-s + (−0.233 − 0.405i)11-s + (0.205 − 0.245i)12-s + (−2.91 − 5.04i)13-s + (−1.49 − 4.12i)15-s − 3.59·16-s + (−1.93 + 3.35i)17-s + (−0.701 − 3.98i)18-s + (1.09 + 1.89i)19-s + ⋯
L(s)  = 1  + 0.952·2-s + (−0.642 + 0.766i)3-s − 0.0923·4-s + (−0.566 + 0.980i)5-s + (−0.612 + 0.729i)6-s − 1.04·8-s + (−0.173 − 0.984i)9-s + (−0.539 + 0.934i)10-s + (−0.0705 − 0.122i)11-s + (0.0593 − 0.0707i)12-s + (−0.807 − 1.39i)13-s + (−0.387 − 1.06i)15-s − 0.899·16-s + (−0.470 + 0.814i)17-s + (−0.165 − 0.938i)18-s + (0.250 + 0.434i)19-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: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0325542 - 0.512986i\)
\(L(\frac12)\) \(\approx\) \(0.0325542 - 0.512986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 - 1.32i)T \)
7 \( 1 \)
good2 \( 1 - 1.34T + 2T^{2} \)
5 \( 1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.233 + 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0530 + 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 + (-3.84 - 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.33T + 47T^{2} \)
53 \( 1 + (-0.358 + 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.736T + 59T^{2} \)
61 \( 1 - 0.958T + 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (-5.13 + 8.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-1.36 + 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.05 - 7.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54426440876305501001428864531, −10.82632602378828751615507395636, −10.07895237226985240375452593975, −9.034530174418036862132225751711, −7.78932261266606445709485794385, −6.62337588031373120156903213480, −5.65399834147712095373816727147, −4.89114962578282056372463793187, −3.68941037594969802968216591028, −3.09697582623663454091332160905, 0.24802698083798666870835540963, 2.26320601607568600418535310076, 4.12173679141026505828035303466, 4.80731246549708467716868329916, 5.62594004901320904889964360985, 6.80262262145254060437597972153, 7.66181958604686024202201860194, 8.881917524908578160484195645292, 9.553861203865325146872309958852, 11.31212711429160931976910102233

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