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Properties

Label 2-912-19.4-c1-0-14
Degree $2$
Conductor $912$
Sign $-0.0534 + 0.998i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (0.173 + 0.984i)3-s + (−1.93 + 1.62i)5-s + (0.266 − 0.460i)7-s + (−0.939 + 0.342i)9-s + (−2.55 − 4.42i)11-s + (0.705 − 4.00i)13-s + (−1.93 − 1.62i)15-s + (−1.82 − 0.664i)17-s + (−2.23 + 3.74i)19-s + (0.5 + 0.181i)21-s + (−2.33 − 1.95i)23-s + (0.245 − 1.39i)25-s + (−0.5 − 0.866i)27-s + (−1.51 + 0.550i)29-s + (4.93 − 8.55i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (−0.867 + 0.727i)5-s + (0.100 − 0.174i)7-s + (−0.313 + 0.114i)9-s + (−0.769 − 1.33i)11-s + (0.195 − 1.11i)13-s + (−0.500 − 0.420i)15-s + (−0.442 − 0.161i)17-s + (−0.512 + 0.858i)19-s + (0.109 + 0.0397i)21-s + (−0.485 − 0.407i)23-s + (0.0490 − 0.278i)25-s + (−0.0962 − 0.166i)27-s + (−0.281 + 0.102i)29-s + (0.887 − 1.53i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0534 + 0.998i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.0534 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415994 - 0.438855i\)
\(L(\frac12)\) \(\approx\) \(0.415994 - 0.438855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (2.23 - 3.74i)T \)
good5 \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.266 + 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.55 + 4.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.705 + 4.00i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.82 + 0.664i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.33 + 1.95i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.51 - 0.550i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.93 + 8.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.10T + 37T^{2} \)
41 \( 1 + (1.47 + 8.34i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.135 + 0.113i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-7.10 + 2.58i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (7.58 + 6.36i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.58 + 1.30i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.6 + 8.95i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (9.59 - 3.49i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.92 - 2.45i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.322 + 1.82i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.82 + 10.3i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.73 - 9.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.02 - 5.83i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.39 - 2.32i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.14518951985726047750423749199, −8.952158387967430656542184812116, −7.974069146251291288160962554354, −7.73998244292472588930559370206, −6.27779979042816335482662818568, −5.58909667572057764194157152663, −4.29468420483364127593914819270, −3.47686801783567870793601811438, −2.63283738872861810648007092098, −0.28009772849151320320362714088, 1.55276527064776908192171223178, 2.72726558111037349015559956817, 4.37312605925283106039698516381, 4.68841416822256704722450716004, 6.12315775846556637405599859588, 7.08788199181698667939447696499, 7.74735804991375292085519718994, 8.601623366930999414517165628058, 9.244143881253726898718695419045, 10.31605078107859519659656121486

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