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Properties

Label 2-42e2-63.59-c1-0-12
Degree $2$
Conductor $1764$
Sign $-0.266 - 0.963i$
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  + (−0.744 + 1.56i)3-s − 0.553·5-s + (−1.89 − 2.32i)9-s + 4.65i·11-s + (3.58 + 2.06i)13-s + (0.412 − 0.866i)15-s + (3.62 − 6.27i)17-s + (5.81 − 3.35i)19-s + 5.60i·23-s − 4.69·25-s + (5.05 − 1.22i)27-s + (1.16 − 0.673i)29-s + (0.830 − 0.479i)31-s + (−7.28 − 3.47i)33-s + (3.53 + 6.12i)37-s + ⋯
L(s)  = 1  + (−0.430 + 0.902i)3-s − 0.247·5-s + (−0.630 − 0.776i)9-s + 1.40i·11-s + (0.993 + 0.573i)13-s + (0.106 − 0.223i)15-s + (0.878 − 1.52i)17-s + (1.33 − 0.770i)19-s + 1.16i·23-s − 0.938·25-s + (0.972 − 0.234i)27-s + (0.216 − 0.125i)29-s + (0.149 − 0.0861i)31-s + (−1.26 − 0.604i)33-s + (0.581 + 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339712891\)
\(L(\frac12)\) \(\approx\) \(1.339712891\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.744 - 1.56i)T \)
7 \( 1 \)
good5 \( 1 + 0.553T + 5T^{2} \)
11 \( 1 - 4.65iT - 11T^{2} \)
13 \( 1 + (-3.58 - 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.62 + 6.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.81 + 3.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.60iT - 23T^{2} \)
29 \( 1 + (-1.16 + 0.673i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 + 0.479i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.53 - 6.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 + 1.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.90 - 8.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.30 + 4.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.89 - 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.37 - 3.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.407iT - 71T^{2} \)
73 \( 1 + (7.47 + 4.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.318 - 0.551i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.78 - 4.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.46 - 6.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.48 + 4.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.792849002725808660247022101089, −9.044052134645071792566918084096, −7.86133987344087423806973715783, −7.18513513616389207072679759678, −6.26149684162373538368480202482, −5.23528794298846437224558447594, −4.70796983465898244938794316959, −3.73528896843515971556426627521, −2.86690903158771458289618640140, −1.20793033416186110115444999176, 0.62981409522322194286450247315, 1.65547490398817297986993509959, 3.14432811803453489033960255392, 3.81173849307506220894617633880, 5.37170005141976519714418753641, 5.90017122010303266632570899403, 6.47551379429808312656454439072, 7.69779555647366664272445019590, 8.165361665088634130564963743693, 8.692978976839819417472205540231

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