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Properties

Label 2-21e2-49.16-c1-0-15
Degree $2$
Conductor $441$
Sign $0.528 - 0.848i$
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  + (0.712 + 1.81i)2-s + (−1.32 + 1.22i)4-s + (2.50 − 1.70i)5-s + (−1.59 − 2.11i)7-s + (0.340 + 0.163i)8-s + (4.87 + 3.32i)10-s + (3.38 + 0.509i)11-s + (2.71 − 3.40i)13-s + (2.70 − 4.39i)14-s + (−0.325 + 4.33i)16-s + (−7.31 + 2.25i)17-s + (0.230 − 0.399i)19-s + (−1.21 + 5.33i)20-s + (1.48 + 6.50i)22-s + (5.60 + 1.72i)23-s + ⋯
L(s)  = 1  + (0.504 + 1.28i)2-s + (−0.662 + 0.614i)4-s + (1.11 − 0.762i)5-s + (−0.601 − 0.798i)7-s + (0.120 + 0.0579i)8-s + (1.54 + 1.05i)10-s + (1.01 + 0.153i)11-s + (0.752 − 0.944i)13-s + (0.722 − 1.17i)14-s + (−0.0812 + 1.08i)16-s + (−1.77 + 0.546i)17-s + (0.0528 − 0.0916i)19-s + (−0.272 + 1.19i)20-s + (0.316 + 1.38i)22-s + (1.16 + 0.360i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.528 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89483 + 1.05206i\)
\(L(\frac12)\) \(\approx\) \(1.89483 + 1.05206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.59 + 2.11i)T \)
good2 \( 1 + (-0.712 - 1.81i)T + (-1.46 + 1.36i)T^{2} \)
5 \( 1 + (-2.50 + 1.70i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-3.38 - 0.509i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (-2.71 + 3.40i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (7.31 - 2.25i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.230 + 0.399i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.60 - 1.72i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (1.51 - 6.63i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.485 + 0.841i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.44 + 4.12i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (5.80 + 2.79i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.08 - 1.48i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.71 + 4.37i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (5.16 - 4.79i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-9.14 - 6.23i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (0.0508 + 0.0472i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-1.88 - 3.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.17 + 5.13i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.98 - 7.59i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-5.82 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.60 + 2.01i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (2.46 - 0.372i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 2.26T + 97T^{2} \)
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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

−11.09630849694471935962822076609, −10.32061774288946219542190495391, −9.092251448569864656469967939965, −8.622986672840131059426518010976, −7.15725646950770479945186974411, −6.56186879355947120378221935555, −5.71627072337468283072702730504, −4.80314872529049373072284965848, −3.68609760724246343431255932905, −1.53213978541588668146720212281, 1.79432097728915395894899197213, 2.64155489228797034512218224119, 3.72245901904417221189564264948, 4.96773193985107700490018315471, 6.47220860865623801523426688820, 6.66460849737180671760523277361, 8.807649867163589209118078314370, 9.422129810578604635295773330001, 10.18863638997051916257674619579, 11.36441156016180654532498794440

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