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Properties

Label 2-21e2-441.110-c1-0-45
Degree $2$
Conductor $441$
Sign $0.834 + 0.550i$
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  + (2.28 − 0.171i)2-s + (0.623 − 1.61i)3-s + (3.23 − 0.487i)4-s + (0.442 + 1.94i)5-s + (1.14 − 3.80i)6-s + (2.56 − 0.639i)7-s + (2.84 − 0.649i)8-s + (−2.22 − 2.01i)9-s + (1.34 + 4.36i)10-s + (−1.84 + 3.83i)11-s + (1.22 − 5.53i)12-s + (−4.17 + 0.313i)13-s + (5.76 − 1.90i)14-s + (3.41 + 0.493i)15-s + (0.149 − 0.0460i)16-s + (1.78 + 0.268i)17-s + ⋯
L(s)  = 1  + (1.61 − 0.121i)2-s + (0.359 − 0.933i)3-s + (1.61 − 0.243i)4-s + (0.198 + 0.867i)5-s + (0.469 − 1.55i)6-s + (0.970 − 0.241i)7-s + (1.00 − 0.229i)8-s + (−0.741 − 0.671i)9-s + (0.425 + 1.38i)10-s + (−0.556 + 1.15i)11-s + (0.354 − 1.59i)12-s + (−1.15 + 0.0868i)13-s + (1.54 − 0.508i)14-s + (0.880 + 0.127i)15-s + (0.0373 − 0.0115i)16-s + (0.432 + 0.0651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.834 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.51753 - 1.05485i\)
\(L(\frac12)\) \(\approx\) \(3.51753 - 1.05485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 + 1.61i)T \)
7 \( 1 + (-2.56 + 0.639i)T \)
good2 \( 1 + (-2.28 + 0.171i)T + (1.97 - 0.298i)T^{2} \)
5 \( 1 + (-0.442 - 1.94i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.84 - 3.83i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.17 - 0.313i)T + (12.8 - 1.93i)T^{2} \)
17 \( 1 + (-1.78 - 0.268i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (5.84 + 3.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.85 + 3.07i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-0.822 + 0.322i)T + (21.2 - 19.7i)T^{2} \)
31 \( 1 + (8.22 + 4.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 5.74i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-7.66 + 7.11i)T + (3.06 - 40.8i)T^{2} \)
43 \( 1 + (-5.11 - 4.74i)T + (3.21 + 42.8i)T^{2} \)
47 \( 1 + (-0.524 - 6.99i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-0.0150 - 0.00589i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (5.74 + 5.33i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (1.68 - 11.1i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (0.481 - 0.834i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.93 + 7.12i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-6.90 + 10.1i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (-4.55 - 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.292 - 3.90i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (0.526 - 7.02i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-0.607 - 0.350i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.21121017165374358621869415265, −10.68397110694669700903157184166, −9.234118488660144574791951959084, −7.75239065578830324919641066547, −7.14013432390782101807817193632, −6.34194056097290939058937157965, −5.12350798009212843016360207213, −4.27264377277371880619905454603, −2.67836101963959008945646088676, −2.18755010495797983878220604729, 2.30010474074204168526520031512, 3.50311392152984415955647445241, 4.57774170298412163521477437784, 5.26391374235296838868115873887, 5.76765943457437726992923378951, 7.50742312142927045670456630245, 8.506494232114476450057694399260, 9.299646686899793048987194439331, 10.70435485534259408229413115072, 11.24085227996499816074523015482

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