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Properties

Label 2-21e2-49.2-c1-0-0
Degree $2$
Conductor $441$
Sign $0.0860 - 0.996i$
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  + (−0.681 − 0.210i)2-s + (−1.23 − 0.840i)4-s + (−2.65 + 0.400i)5-s + (−0.518 − 2.59i)7-s + (1.55 + 1.94i)8-s + (1.89 + 0.285i)10-s + (1.45 − 1.34i)11-s + (0.408 + 1.78i)13-s + (−0.191 + 1.87i)14-s + (0.442 + 1.12i)16-s + (0.176 + 2.35i)17-s + (−3.23 + 5.60i)19-s + (3.61 + 1.74i)20-s + (−1.27 + 0.612i)22-s + (−0.536 + 7.15i)23-s + ⋯
L(s)  = 1  + (−0.481 − 0.148i)2-s + (−0.616 − 0.420i)4-s + (−1.18 + 0.179i)5-s + (−0.196 − 0.980i)7-s + (0.548 + 0.687i)8-s + (0.599 + 0.0903i)10-s + (0.438 − 0.406i)11-s + (0.113 + 0.495i)13-s + (−0.0512 + 0.501i)14-s + (0.110 + 0.281i)16-s + (0.0427 + 0.571i)17-s + (−0.742 + 1.28i)19-s + (0.808 + 0.389i)20-s + (−0.271 + 0.130i)22-s + (−0.111 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0860 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0860 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0860 - 0.996i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0860 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275478 + 0.252698i\)
\(L(\frac12)\) \(\approx\) \(0.275478 + 0.252698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.518 + 2.59i)T \)
good2 \( 1 + (0.681 + 0.210i)T + (1.65 + 1.12i)T^{2} \)
5 \( 1 + (2.65 - 0.400i)T + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (-1.45 + 1.34i)T + (0.822 - 10.9i)T^{2} \)
13 \( 1 + (-0.408 - 1.78i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.176 - 2.35i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (3.23 - 5.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.536 - 7.15i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (2.13 + 1.02i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-2.98 - 5.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.06 - 0.724i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-1.15 - 1.44i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-6.12 + 7.67i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (4.45 + 1.37i)T + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (-0.302 - 0.206i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-4.96 - 0.748i)T + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (-0.00167 + 0.00113i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (-5.23 - 9.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.12 - 3.43i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.23 - 0.380i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (0.184 - 0.319i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.44 - 15.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (4.91 + 4.56i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + 5.61T + 97T^{2} \)
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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

−11.15952213082247069048765973543, −10.43876765343976878548647381010, −9.622270431064014190348864253626, −8.521725444208700507411416654754, −7.88608579104069268156804382654, −6.89386366764173461110248226821, −5.64296145441993197301270504926, −4.14292717738439332059535572958, −3.73923902849412412525462643960, −1.37321201594970829410901520950, 0.31048229209757993681129040416, 2.77146265300746240487809758526, 4.10890804894120415546605086558, 4.86516167771446964868387285774, 6.41683476008190574077469656349, 7.47739874875209483410535997594, 8.301263281419290182429408970200, 8.933399246565752785127967510305, 9.722283942375992254530884428559, 10.98025036395591587062787235469

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