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Properties

Label 2-882-441.142-c1-0-7
Degree $2$
Conductor $882$
Sign $0.0386 - 0.999i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
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  + (−0.365 + 0.930i)2-s + (−0.323 − 1.70i)3-s + (−0.733 − 0.680i)4-s + (−1.74 + 0.842i)5-s + (1.70 + 0.320i)6-s + (−1.87 − 1.86i)7-s + (0.900 − 0.433i)8-s + (−2.79 + 1.10i)9-s + (−0.145 − 1.93i)10-s + (0.351 + 0.440i)11-s + (−0.919 + 1.46i)12-s + (−0.244 + 0.623i)13-s + (2.42 − 1.06i)14-s + (1.99 + 2.70i)15-s + (0.0747 + 0.997i)16-s + (4.07 − 3.78i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (−0.187 − 0.982i)3-s + (−0.366 − 0.340i)4-s + (−0.782 + 0.376i)5-s + (0.694 + 0.130i)6-s + (−0.709 − 0.705i)7-s + (0.318 − 0.153i)8-s + (−0.930 + 0.367i)9-s + (−0.0458 − 0.612i)10-s + (0.106 + 0.132i)11-s + (−0.265 + 0.423i)12-s + (−0.0678 + 0.172i)13-s + (0.647 − 0.284i)14-s + (0.516 + 0.697i)15-s + (0.0186 + 0.249i)16-s + (0.989 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0386 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0386 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0386 - 0.999i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.0386 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404181 + 0.388868i\)
\(L(\frac12)\) \(\approx\) \(0.404181 + 0.388868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.365 - 0.930i)T \)
3 \( 1 + (0.323 + 1.70i)T \)
7 \( 1 + (1.87 + 1.86i)T \)
good5 \( 1 + (1.74 - 0.842i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.351 - 0.440i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.244 - 0.623i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-4.07 + 3.78i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.59 - 4.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.562 + 2.46i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-3.83 - 1.18i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (5.44 - 9.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.91 - 1.20i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-3.12 - 2.13i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-5.18 + 3.53i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (3.26 - 8.31i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-7.09 + 2.18i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (1.82 - 1.24i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (0.284 - 0.264i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (3.30 - 5.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.13 - 9.36i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-13.0 - 1.97i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (3.43 + 5.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.43 - 3.64i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (1.09 + 2.79i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.94 - 6.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36551846184803385711088851778, −9.384000674953075534600865100770, −8.342035539357835169422243548738, −7.57353871117591186232491229519, −7.05820018381044645192382848624, −6.35044980069078570952657905545, −5.33505346665033572313243382744, −4.02446764406742371458699350768, −2.93349720082665271364302304117, −1.11509244898279952977488981797, 0.35857488909828193428263886408, 2.51517729781515956684212219975, 3.59503587240756394463627469402, 4.25352416613923965974022247074, 5.39784321731479505447303561504, 6.24225718569224347723125840418, 7.73331315742779321652030109719, 8.519952132083254339379587191310, 9.269076793305325614197829789693, 9.865787412066787498254687441624

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