Properties

Label 2-882-7.2-c3-0-32
Degree $2$
Conductor $882$
Sign $0.386 + 0.922i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (4.69 − 8.12i)5-s − 7.99·8-s + (−9.38 − 16.2i)10-s + (10 + 17.3i)11-s + 65.6·13-s + (−8 + 13.8i)16-s + (28.1 + 48.7i)17-s + (−4.69 + 8.12i)19-s − 37.5·20-s + 40·22-s + (24 − 41.5i)23-s + (18.4 + 32.0i)25-s + (65.6 − 113. i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.419 − 0.726i)5-s − 0.353·8-s + (−0.296 − 0.513i)10-s + (0.274 + 0.474i)11-s + 1.40·13-s + (−0.125 + 0.216i)16-s + (0.401 + 0.695i)17-s + (−0.0566 + 0.0980i)19-s − 0.419·20-s + 0.387·22-s + (0.217 − 0.376i)23-s + (0.147 + 0.256i)25-s + (0.495 − 0.857i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.021416236\)
\(L(\frac12)\) \(\approx\) \(3.021416236\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-4.69 + 8.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-10 - 17.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 65.6T + 2.19e3T^{2} \)
17 \( 1 + (-28.1 - 48.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (4.69 - 8.12i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-24 + 41.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 166T + 2.43e4T^{2} \)
31 \( 1 + (-103. - 178. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-39 + 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 436T + 7.95e4T^{2} \)
47 \( 1 + (-103. + 178. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-31 - 53.6i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (333. + 576. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (136. - 235. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (290 + 502. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 544T + 3.57e5T^{2} \)
73 \( 1 + (-300. - 519. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-340 + 588. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 196.T + 5.71e5T^{2} \)
89 \( 1 + (750. - 1.29e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 656.T + 9.12e5T^{2} \)
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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

−9.666445111010474082636427685855, −8.805145984705185862547843367784, −8.239609815589365251514219739303, −6.79493554135649857650401334625, −5.96460568717282780433992665691, −5.05964019199122690336129594300, −4.15774273458987243670903734212, −3.16737201111217695423377132048, −1.75130485005546348402793536194, −0.960557840292143462694534675432, 0.977565322822245351735779566444, 2.65407077185151287912117324429, 3.53829197894345859521657427419, 4.62507034621758651833025703653, 5.84505398010262154759782944284, 6.31273507305632200413145822321, 7.18630566665167139881117422507, 8.171547400207640199098954479125, 8.923769625815802997246693387415, 9.870773514556188989096144141211

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