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Properties

Label 2-387-9.4-c1-0-24
Degree $2$
Conductor $387$
Sign $0.988 - 0.148i$
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  + (0.649 + 1.12i)2-s + (−1.24 + 1.20i)3-s + (0.156 − 0.271i)4-s + (1.38 − 2.39i)5-s + (−2.16 − 0.609i)6-s + (−0.516 − 0.894i)7-s + 3.00·8-s + (0.0766 − 2.99i)9-s + 3.58·10-s + (−1.47 − 2.54i)11-s + (0.133 + 0.526i)12-s + (1.68 − 2.91i)13-s + (0.670 − 1.16i)14-s + (1.17 + 4.63i)15-s + (1.63 + 2.83i)16-s + 1.55·17-s + ⋯
L(s)  = 1  + (0.459 + 0.795i)2-s + (−0.716 + 0.698i)3-s + (0.0783 − 0.135i)4-s + (0.617 − 1.07i)5-s + (−0.883 − 0.248i)6-s + (−0.195 − 0.337i)7-s + 1.06·8-s + (0.0255 − 0.999i)9-s + 1.13·10-s + (−0.443 − 0.768i)11-s + (0.0386 + 0.151i)12-s + (0.467 − 0.809i)13-s + (0.179 − 0.310i)14-s + (0.304 + 1.19i)15-s + (0.409 + 0.708i)16-s + 0.376·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.988 - 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61474 + 0.120498i\)
\(L(\frac12)\) \(\approx\) \(1.61474 + 0.120498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 - 1.20i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.649 - 1.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.38 + 2.39i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.516 + 0.894i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.47 + 2.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 - 0.820T + 19T^{2} \)
23 \( 1 + (4.29 - 7.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.623 - 1.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.962 + 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.44T + 37T^{2} \)
41 \( 1 + (3.13 - 5.43i)T + (-20.5 - 35.5i)T^{2} \)
47 \( 1 + (-5.05 - 8.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.79T + 53T^{2} \)
59 \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0348 - 0.0603i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.76 + 9.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.88T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + (3.49 + 6.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.50 + 2.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + (-1.93 - 3.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.23072452849932497553244281264, −10.35516814084574897937207707756, −9.699578487648956725737326531849, −8.556572734931213243393475014267, −7.44288123409179598586776605327, −6.04570220127663494705968354577, −5.63071002044051374889760693612, −4.88085149976390499734076103063, −3.62299110808050132774373243564, −1.13233830426488051923761362678, 1.93108936140122277462155583628, 2.67956173900463772054298965832, 4.22140984969990932996086950588, 5.53144301390524958582132505025, 6.64658681650421667360857960033, 7.19062792281245629546685732336, 8.427913413438916720256932985987, 10.14052650950341215720007589656, 10.47461445068506202205570990451, 11.46235201534620428102837418079

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