Properties

Label 2-387-1.1-c3-0-46
Degree $2$
Conductor $387$
Sign $-1$
Analytic cond. $22.8337$
Root an. cond. $4.77846$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.15·2-s + 1.96·4-s − 1.36·5-s + 13.0·7-s − 19.0·8-s − 4.30·10-s − 64.7·11-s − 19.2·13-s + 41.0·14-s − 75.8·16-s + 54.1·17-s − 69.0·19-s − 2.67·20-s − 204.·22-s − 29.6·23-s − 123.·25-s − 60.9·26-s + 25.5·28-s − 13.1·29-s + 185.·31-s − 87.0·32-s + 170.·34-s − 17.7·35-s − 369.·37-s − 218.·38-s + 25.9·40-s + 294.·41-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.245·4-s − 0.121·5-s + 0.702·7-s − 0.842·8-s − 0.136·10-s − 1.77·11-s − 0.411·13-s + 0.784·14-s − 1.18·16-s + 0.772·17-s − 0.833·19-s − 0.0299·20-s − 1.98·22-s − 0.268·23-s − 0.985·25-s − 0.459·26-s + 0.172·28-s − 0.0840·29-s + 1.07·31-s − 0.480·32-s + 0.861·34-s − 0.0857·35-s − 1.64·37-s − 0.930·38-s + 0.102·40-s + 1.12·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(22.8337\)
Root analytic conductor: \(4.77846\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 387,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 43T \)
good2 \( 1 - 3.15T + 8T^{2} \)
5 \( 1 + 1.36T + 125T^{2} \)
7 \( 1 - 13.0T + 343T^{2} \)
11 \( 1 + 64.7T + 1.33e3T^{2} \)
13 \( 1 + 19.2T + 2.19e3T^{2} \)
17 \( 1 - 54.1T + 4.91e3T^{2} \)
19 \( 1 + 69.0T + 6.85e3T^{2} \)
23 \( 1 + 29.6T + 1.21e4T^{2} \)
29 \( 1 + 13.1T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 + 369.T + 5.06e4T^{2} \)
41 \( 1 - 294.T + 6.89e4T^{2} \)
47 \( 1 + 367.T + 1.03e5T^{2} \)
53 \( 1 + 708.T + 1.48e5T^{2} \)
59 \( 1 + 116.T + 2.05e5T^{2} \)
61 \( 1 - 218.T + 2.26e5T^{2} \)
67 \( 1 + 133.T + 3.00e5T^{2} \)
71 \( 1 - 926.T + 3.57e5T^{2} \)
73 \( 1 - 455.T + 3.89e5T^{2} \)
79 \( 1 + 620.T + 4.93e5T^{2} \)
83 \( 1 - 1.31e3T + 5.71e5T^{2} \)
89 \( 1 + 509.T + 7.04e5T^{2} \)
97 \( 1 - 965.T + 9.12e5T^{2} \)
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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

−10.60512229301919488667779692919, −9.685776828962370226345696634049, −8.320449673406720848490907813222, −7.71988456766223119369219631848, −6.28317489401454061446700187726, −5.24830676940554805169476690130, −4.68542259197288803088603090618, −3.39783154111978565925044862582, −2.22891666055076259276258524745, 0, 2.22891666055076259276258524745, 3.39783154111978565925044862582, 4.68542259197288803088603090618, 5.24830676940554805169476690130, 6.28317489401454061446700187726, 7.71988456766223119369219631848, 8.320449673406720848490907813222, 9.685776828962370226345696634049, 10.60512229301919488667779692919

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