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Properties

Label 2-882-147.101-c1-0-4
Degree $2$
Conductor $882$
Sign $-0.363 - 0.931i$
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  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (2.38 + 1.62i)5-s + (−1.53 + 2.15i)7-s + (−0.433 − 0.900i)8-s + (−1.62 − 2.38i)10-s + (0.328 + 2.18i)11-s + (−1.90 + 1.51i)13-s + (2.21 − 1.44i)14-s + (0.0747 + 0.997i)16-s + (1.54 + 0.478i)17-s + (−5.96 + 3.44i)19-s + (0.642 + 2.81i)20-s + (0.490 − 2.15i)22-s + (−0.270 − 0.878i)23-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (0.366 + 0.340i)4-s + (1.06 + 0.727i)5-s + (−0.578 + 0.815i)7-s + (−0.153 − 0.318i)8-s + (−0.514 − 0.754i)10-s + (0.0991 + 0.657i)11-s + (−0.527 + 0.420i)13-s + (0.591 − 0.387i)14-s + (0.0186 + 0.249i)16-s + (0.375 + 0.115i)17-s + (−1.36 + 0.789i)19-s + (0.143 + 0.629i)20-s + (0.104 − 0.458i)22-s + (−0.0564 − 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.363 - 0.931i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.363 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533209 + 0.780807i\)
\(L(\frac12)\) \(\approx\) \(0.533209 + 0.780807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.930 + 0.365i)T \)
3 \( 1 \)
7 \( 1 + (1.53 - 2.15i)T \)
good5 \( 1 + (-2.38 - 1.62i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.328 - 2.18i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (1.90 - 1.51i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.54 - 0.478i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (5.96 - 3.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.270 + 0.878i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-3.46 + 0.790i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (7.19 + 4.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.79 - 2.59i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (1.04 - 0.502i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (0.507 + 0.244i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.28 + 5.81i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (6.70 - 7.22i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-4.53 + 3.08i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-4.79 - 5.16i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (1.18 - 2.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.34 - 0.763i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-9.86 + 3.87i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-8.54 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.46 - 9.36i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-7.83 - 1.18i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 9.32iT - 97T^{2} \)
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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

−10.16440164332170058074822570611, −9.697750495107223857405525051198, −8.960920123741544670897379054997, −7.986887526809222784177109614721, −6.81370845126088608599070446089, −6.32684955764239665313316868569, −5.35088475289296483054337217959, −3.84702441072964769095265012545, −2.51183309471271169490878832146, −1.95451330390134124112544041702, 0.52835311670964778811499895549, 1.88936401693796948268975191413, 3.27531430111781058353164210971, 4.72247697230329878481982945851, 5.64912131433432523008132040661, 6.48667897472791567585030238486, 7.29177796743331108861868145607, 8.384264631174616700371118267840, 9.091428167295434367521222016229, 9.779810867000127062090287281853

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