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Properties

Label 2-882-63.38-c1-0-29
Degree $2$
Conductor $882$
Sign $0.283 + 0.958i$
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  − i·2-s + (1.70 − 0.315i)3-s − 4-s + (−0.220 + 0.381i)5-s + (−0.315 − 1.70i)6-s + i·8-s + (2.80 − 1.07i)9-s + (0.381 + 0.220i)10-s + (0.450 − 0.260i)11-s + (−1.70 + 0.315i)12-s + (5.55 − 3.20i)13-s + (−0.254 + 0.718i)15-s + 16-s + (0.163 − 0.283i)17-s + (−1.07 − 2.80i)18-s + (−3.67 + 2.11i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.983 − 0.182i)3-s − 0.5·4-s + (−0.0984 + 0.170i)5-s + (−0.128 − 0.695i)6-s + 0.353i·8-s + (0.933 − 0.358i)9-s + (0.120 + 0.0695i)10-s + (0.135 − 0.0784i)11-s + (−0.491 + 0.0911i)12-s + (1.53 − 0.888i)13-s + (−0.0657 + 0.185i)15-s + 0.250·16-s + (0.0396 − 0.0687i)17-s + (−0.253 − 0.660i)18-s + (−0.842 + 0.486i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.283 + 0.958i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76420 - 1.31772i\)
\(L(\frac12)\) \(\approx\) \(1.76420 - 1.31772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.70 + 0.315i)T \)
7 \( 1 \)
good5 \( 1 + (0.220 - 0.381i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.450 + 0.260i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.55 + 3.20i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.163 + 0.283i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 - 2.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.25 + 0.725i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.74 - 3.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.01iT - 31T^{2} \)
37 \( 1 + (1.84 + 3.18i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.96 + 5.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.05T + 47T^{2} \)
53 \( 1 + (-6.89 - 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.90T + 59T^{2} \)
61 \( 1 - 1.54iT - 61T^{2} \)
67 \( 1 - 6.52T + 67T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 + (3.57 + 2.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.32T + 79T^{2} \)
83 \( 1 + (8.55 - 14.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.86 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.6 - 6.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12702262765944237583551261512, −8.932815805123602009079168019259, −8.547010315841794936085558596227, −7.67979740400769270030307732545, −6.59802836678390709469668794431, −5.52579530892247085225634360402, −4.07626152886156457360990487724, −3.48726939576722375127426975882, −2.41505278598791604921235298318, −1.15451346028770130236130315007, 1.49953982245840927099602800434, 3.03912460342406737469915865123, 4.14494208729413194863165063668, 4.77008452780413518096274392033, 6.32120033356691075901028377002, 6.76987825367854531706068651836, 8.070182046058192795662143871894, 8.504480598590558289312163907498, 9.166004171682733618778594667099, 10.08775241636352760345413819821

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