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Properties

Label 2-21e2-49.23-c1-0-18
Degree $2$
Conductor $441$
Sign $-0.687 + 0.726i$
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  + (0.0474 − 0.633i)2-s + (1.57 + 0.237i)4-s + (−1.59 − 1.47i)5-s + (−1.92 − 1.81i)7-s + (0.508 − 2.22i)8-s + (−1.01 + 0.937i)10-s + (−3.64 − 2.48i)11-s + (−2.93 + 1.41i)13-s + (−1.24 + 1.13i)14-s + (1.66 + 0.513i)16-s + (0.736 − 1.87i)17-s + (2.06 − 3.57i)19-s + (−2.15 − 2.70i)20-s + (−1.74 + 2.19i)22-s + (2.47 + 6.30i)23-s + ⋯
L(s)  = 1  + (0.0335 − 0.447i)2-s + (0.789 + 0.118i)4-s + (−0.711 − 0.659i)5-s + (−0.726 − 0.687i)7-s + (0.179 − 0.787i)8-s + (−0.319 + 0.296i)10-s + (−1.10 − 0.750i)11-s + (−0.813 + 0.391i)13-s + (−0.332 + 0.302i)14-s + (0.416 + 0.128i)16-s + (0.178 − 0.455i)17-s + (0.474 − 0.821i)19-s + (−0.482 − 0.605i)20-s + (−0.372 + 0.467i)22-s + (0.516 + 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.687 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439449 - 1.02054i\)
\(L(\frac12)\) \(\approx\) \(0.439449 - 1.02054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 + 1.81i)T \)
good2 \( 1 + (-0.0474 + 0.633i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (1.59 + 1.47i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (3.64 + 2.48i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (2.93 - 1.41i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.736 + 1.87i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-2.06 + 3.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.47 - 6.30i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-2.38 - 2.99i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (5.13 + 8.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.25 + 1.39i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (-1.17 + 5.13i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.437 - 1.91i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (0.630 - 8.40i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (1.39 + 0.209i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (-3.33 + 3.08i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (4.98 - 0.750i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (2.14 + 3.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.90 + 9.91i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.542 - 7.24i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-2.19 + 3.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.51 + 1.21i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (4.98 - 3.39i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 - 1.29T + 97T^{2} \)
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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

−11.07597269231790934090701113730, −9.951638425554423035084758160444, −9.192120259059679958907850093416, −7.62833967169939765491491643425, −7.46295326682945510192710700145, −6.09594785713165140094021447872, −4.82367906880707901102955694698, −3.60575226996410974058965030219, −2.63509981933815753452037186788, −0.65264217264868865349273233493, 2.38595374898674743104777214265, 3.21464880680955995184774112414, 4.94502933212124364066946381513, 5.93218675368192019137411302043, 6.94144632172509980868915592116, 7.58066230624821922801581381607, 8.436178962140549172053194379817, 9.938009184574720486709870614979, 10.46729460742385129183778509397, 11.45877419750945328876781226452

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