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Properties

Label 2-21e2-63.20-c1-0-17
Degree $2$
Conductor $441$
Sign $0.947 - 0.319i$
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  + (1.58 + 0.916i)2-s + (−1.44 − 0.958i)3-s + (0.678 + 1.17i)4-s + (0.322 + 0.559i)5-s + (−1.41 − 2.84i)6-s − 1.17i·8-s + (1.16 + 2.76i)9-s + 1.18i·10-s + (4.60 + 2.65i)11-s + (0.147 − 2.34i)12-s + (4.44 − 2.56i)13-s + (0.0702 − 1.11i)15-s + (2.43 − 4.22i)16-s + 1.62·17-s + (−0.689 + 5.45i)18-s + 2.41i·19-s + ⋯
L(s)  = 1  + (1.12 + 0.647i)2-s + (−0.832 − 0.553i)3-s + (0.339 + 0.587i)4-s + (0.144 + 0.250i)5-s + (−0.575 − 1.16i)6-s − 0.416i·8-s + (0.387 + 0.921i)9-s + 0.374i·10-s + (1.38 + 0.801i)11-s + (0.0425 − 0.676i)12-s + (1.23 − 0.711i)13-s + (0.0181 − 0.288i)15-s + (0.609 − 1.05i)16-s + 0.395·17-s + (−0.162 + 1.28i)18-s + 0.553i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08295 + 0.341997i\)
\(L(\frac12)\) \(\approx\) \(2.08295 + 0.341997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 + 0.958i)T \)
7 \( 1 \)
good2 \( 1 + (-1.58 - 0.916i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.322 - 0.559i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.60 - 2.65i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.44 + 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 - 2.41iT - 19T^{2} \)
23 \( 1 + (1.27 - 0.735i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.43 + 3.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.90 - 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + (-5.99 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.54 - 2.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.36iT - 53T^{2} \)
59 \( 1 + (-1.47 - 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.18 + 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.51 + 6.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.32T + 89T^{2} \)
97 \( 1 + (14.3 + 8.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.44444481573668478969229474018, −10.45268040287688142102866231264, −9.481722896884447781117703990917, −8.002224968581356848335221775208, −7.02864547576112265716626749957, −6.26539178647323746608622435333, −5.70209541854751331117551943055, −4.54384995339210251257550521513, −3.54656491060453892390035281009, −1.44065708242422792746051915353, 1.49226825091914737171489861006, 3.59590875664951604602301186911, 3.95302092293859798713185022129, 5.26207194387663141429302696269, 5.91760652379260241267084214130, 6.92947723803561905599597389202, 8.776610195341250631127901393355, 9.249058908292403401332689111339, 10.79452699391960909081467099441, 11.16015208338876080693335971210

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