Properties

Label 2-387-43.14-c1-0-10
Degree $2$
Conductor $387$
Sign $0.874 + 0.484i$
Analytic cond. $3.09021$
Root an. cond. $1.75789$
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  + (−0.993 + 0.478i)2-s + (−0.488 + 0.612i)4-s + (3.17 − 0.979i)5-s + (1.23 − 2.13i)7-s + (0.683 − 2.99i)8-s + (−2.68 + 2.49i)10-s + (−0.748 − 0.937i)11-s + (−4.22 − 3.91i)13-s + (−0.203 + 2.71i)14-s + (0.404 + 1.77i)16-s + (−1.02 − 0.314i)17-s + (2.13 + 0.321i)19-s + (−0.951 + 2.42i)20-s + (1.19 + 0.574i)22-s + (2.16 − 5.50i)23-s + ⋯
L(s)  = 1  + (−0.702 + 0.338i)2-s + (−0.244 + 0.306i)4-s + (1.41 − 0.437i)5-s + (0.465 − 0.807i)7-s + (0.241 − 1.05i)8-s + (−0.849 + 0.788i)10-s + (−0.225 − 0.282i)11-s + (−1.17 − 1.08i)13-s + (−0.0542 + 0.724i)14-s + (0.101 + 0.442i)16-s + (−0.247 − 0.0763i)17-s + (0.489 + 0.0737i)19-s + (−0.212 + 0.542i)20-s + (0.254 + 0.122i)22-s + (0.450 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02594 - 0.265422i\)
\(L(\frac12)\) \(\approx\) \(1.02594 - 0.265422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (2.45 - 6.08i)T \)
good2 \( 1 + (0.993 - 0.478i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-3.17 + 0.979i)T + (4.13 - 2.81i)T^{2} \)
7 \( 1 + (-1.23 + 2.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.748 + 0.937i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.22 + 3.91i)T + (0.971 + 12.9i)T^{2} \)
17 \( 1 + (1.02 + 0.314i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-2.13 - 0.321i)T + (18.1 + 5.60i)T^{2} \)
23 \( 1 + (-2.16 + 5.50i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (0.177 - 2.37i)T + (-28.6 - 4.32i)T^{2} \)
31 \( 1 + (-6.31 - 4.30i)T + (11.3 + 28.8i)T^{2} \)
37 \( 1 + (2.83 + 4.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.77 + 4.22i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (3.30 - 4.14i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.69 - 2.50i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (0.208 + 0.914i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.31 - 1.57i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (-11.8 - 1.78i)T + (64.0 + 19.7i)T^{2} \)
71 \( 1 + (-5.54 - 14.1i)T + (-52.0 + 48.2i)T^{2} \)
73 \( 1 + (4.99 + 4.63i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (1.18 - 2.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.533 - 7.11i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-0.113 - 1.51i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (6.73 + 8.44i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.82533963416598778379036894615, −10.11133131131109717433712005140, −9.455425928275535722140863340088, −8.502466430775812664435680023849, −7.65768993556091829794769332515, −6.71842859527599256761327063556, −5.42046880445073162858956936548, −4.51364956285583305519081785731, −2.77911934599765919510046615248, −0.957278861119717071059756055013, 1.78470756201342889083503536915, 2.50092301588564127909044915326, 4.81104423378735217183414485634, 5.52930338319139233245086309227, 6.60804254386900245069437556422, 7.87601157774507959295956466699, 9.078563262452401811723960934391, 9.607980799275257927538045461333, 10.15408451685606140638866835206, 11.29297109563291839337604550499

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