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Properties

Label 2-42e2-12.11-c1-0-80
Degree $2$
Conductor $1764$
Sign $-0.775 - 0.630i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.03 − 0.965i)2-s + (0.133 − 1.99i)4-s − 2.73i·5-s + (−1.78 − 2.19i)8-s + (−2.63 − 2.82i)10-s − 5.64·11-s + 1.41·13-s + (−3.96 − 0.534i)16-s + 6.19i·17-s − 4.13i·19-s + (−5.45 − 0.366i)20-s + (−5.83 + 5.45i)22-s − 5.64·23-s − 2.46·25-s + (1.46 − 1.36i)26-s + ⋯
L(s)  = 1  + (0.730 − 0.683i)2-s + (0.0669 − 0.997i)4-s − 1.22i·5-s + (−0.632 − 0.774i)8-s + (−0.834 − 0.892i)10-s − 1.70·11-s + 0.392·13-s + (−0.991 − 0.133i)16-s + 1.50i·17-s − 0.947i·19-s + (−1.21 − 0.0818i)20-s + (−1.24 + 1.16i)22-s − 1.17·23-s − 0.492·25-s + (0.286 − 0.267i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.775 - 0.630i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.775 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248781815\)
\(L(\frac12)\) \(\approx\) \(1.248781815\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 + 0.965i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.73iT - 5T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 6.19iT - 17T^{2} \)
19 \( 1 + 4.13iT - 19T^{2} \)
23 \( 1 + 5.64T + 23T^{2} \)
29 \( 1 - 0.378iT - 29T^{2} \)
31 \( 1 + 7.15iT - 31T^{2} \)
37 \( 1 + 3.46T + 37T^{2} \)
41 \( 1 - 5.26iT - 41T^{2} \)
43 \( 1 + 5.84iT - 43T^{2} \)
47 \( 1 - 2.13T + 47T^{2} \)
53 \( 1 + 0.656iT - 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 - 7.98iT - 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 13.8iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 5.26iT - 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−8.707194151865405294320753353953, −8.329695766778987938188012527521, −7.20013160538863066416078207326, −5.90588732998077999223445880407, −5.50899766775989187160461716389, −4.58105934676865268869777196187, −3.92281716428112756297803981289, −2.66470885083020159344563422905, −1.69937797511483983292399138004, −0.33478897416353256759110442435, 2.37552670981771655320296593424, 3.01841499803379933425104412915, 3.92138391228790749366588594206, 5.12478026649806729279523526781, 5.67661043949771874273713837152, 6.65948565652704565855210194688, 7.28386980956820837233840878682, 7.922220565335898713415351783900, 8.684311449079222781976790369574, 9.966349426730359077242548345246

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