Properties

Label 2-21e2-441.142-c1-0-51
Degree $2$
Conductor $441$
Sign $-0.809 + 0.586i$
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  + (1.00 − 2.54i)2-s + (1.48 + 0.897i)3-s + (−4.02 − 3.73i)4-s + (3.08 − 1.48i)5-s + (3.77 − 2.87i)6-s + (−1.19 − 2.35i)7-s + (−8.62 + 4.15i)8-s + (1.38 + 2.65i)9-s + (−0.701 − 9.36i)10-s + (1.41 + 1.76i)11-s + (−2.61 − 9.15i)12-s + (−1.60 + 4.10i)13-s + (−7.21 + 0.696i)14-s + (5.91 + 0.570i)15-s + (1.13 + 15.1i)16-s + (−1.69 + 1.57i)17-s + ⋯
L(s)  = 1  + (0.707 − 1.80i)2-s + (0.855 + 0.518i)3-s + (−2.01 − 1.86i)4-s + (1.38 − 0.665i)5-s + (1.53 − 1.17i)6-s + (−0.453 − 0.891i)7-s + (−3.05 + 1.46i)8-s + (0.462 + 0.886i)9-s + (−0.221 − 2.96i)10-s + (0.425 + 0.533i)11-s + (−0.753 − 2.64i)12-s + (−0.446 + 1.13i)13-s + (−1.92 + 0.186i)14-s + (1.52 + 0.147i)15-s + (0.284 + 3.79i)16-s + (−0.411 + 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.809 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801346 - 2.47121i\)
\(L(\frac12)\) \(\approx\) \(0.801346 - 2.47121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.48 - 0.897i)T \)
7 \( 1 + (1.19 + 2.35i)T \)
good2 \( 1 + (-1.00 + 2.54i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-3.08 + 1.48i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.41 - 1.76i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.60 - 4.10i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (1.69 - 1.57i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.93 + 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.667 - 2.92i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.41 + 0.437i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (1.06 - 1.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.96 + 0.914i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (2.52 + 1.71i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (1.19 - 0.817i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.41 + 3.60i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-9.53 + 2.94i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-3.66 + 2.49i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (7.74 - 7.18i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-5.54 + 9.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.45 - 6.38i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-6.13 - 0.924i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-4.22 - 7.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.508 - 1.29i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (5.49 + 13.9i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (2.34 - 4.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60594555109667213221296445150, −9.865148050210985288295204222604, −9.401643110026187657711410924573, −8.864188318025811645416219534348, −6.86668803180482212136147478146, −5.33907810017951919812715396261, −4.54788305444727181095298965794, −3.70028234604959601884368403912, −2.37715074109915597492028607759, −1.51929640664091882827727538742, 2.63245175586819366224254878306, 3.54692819889811445820901078058, 5.32370052622025518174667723206, 6.02354454347596056106177738937, 6.63354892055432747305678515359, 7.61073494151471878251405892614, 8.528404878336746852145909707346, 9.299731074509831451345714526263, 10.03109498949272561334973818701, 12.10847292812281215076386049980

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