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Properties

Label 2-387-9.7-c1-0-29
Degree $2$
Conductor $387$
Sign $0.974 + 0.226i$
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  + (−1.04 + 1.80i)2-s + (1.25 − 1.19i)3-s + (−1.18 − 2.05i)4-s + (−0.136 − 0.236i)5-s + (0.842 + 3.52i)6-s + (1.21 − 2.09i)7-s + 0.767·8-s + (0.161 − 2.99i)9-s + 0.571·10-s + (0.799 − 1.38i)11-s + (−3.93 − 1.16i)12-s + (−1.73 − 3.00i)13-s + (2.52 + 4.38i)14-s + (−0.453 − 0.134i)15-s + (1.56 − 2.71i)16-s − 3.58·17-s + ⋯
L(s)  = 1  + (−0.738 + 1.27i)2-s + (0.725 − 0.687i)3-s + (−0.591 − 1.02i)4-s + (−0.0611 − 0.105i)5-s + (0.344 + 1.43i)6-s + (0.457 − 0.792i)7-s + 0.271·8-s + (0.0536 − 0.998i)9-s + 0.180·10-s + (0.241 − 0.417i)11-s + (−1.13 − 0.336i)12-s + (−0.480 − 0.832i)13-s + (0.675 + 1.17i)14-s + (−0.117 − 0.0347i)15-s + (0.391 − 0.677i)16-s − 0.869·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07757 - 0.123505i\)
\(L(\frac12)\) \(\approx\) \(1.07757 - 0.123505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 + 1.19i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.04 - 1.80i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.136 + 0.236i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.21 + 2.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.799 + 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.73 + 3.00i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 - 0.324T + 19T^{2} \)
23 \( 1 + (-2.17 - 3.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.950 - 1.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.23T + 37T^{2} \)
41 \( 1 + (-5.20 - 9.02i)T + (-20.5 + 35.5i)T^{2} \)
47 \( 1 + (4.35 - 7.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.52T + 53T^{2} \)
59 \( 1 + (4.95 + 8.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.50 - 9.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.86 - 4.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.74T + 71T^{2} \)
73 \( 1 - 0.832T + 73T^{2} \)
79 \( 1 + (-4.00 + 6.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.40 + 5.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.69T + 89T^{2} \)
97 \( 1 + (0.848 - 1.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19719668689967847437149522567, −9.962576312832700609123433823246, −9.105792718877824552879801408679, −8.181513247366333085207572176838, −7.72325995118513544466623424511, −6.83993279220533741856169602425, −6.00392544645187010939862048122, −4.52978740444497754088948830215, −2.91082956893655633566227808148, −0.883132581333481685690647598901, 1.92128123730653236646201679013, 2.73969310478640626129017173124, 4.04228625467047324354692364983, 5.12539002844857544966699068649, 6.87492637170104873273753105000, 8.224990423825062040920845940284, 9.049103982553195845717186624838, 9.377365693328636184145569464298, 10.51333821041021239986376648071, 11.10812054643824771070016848878

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