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Properties

Label 2-387-43.42-c4-0-60
Degree $2$
Conductor $387$
Sign $-0.0745 + 0.997i$
Analytic cond. $40.0041$
Root an. cond. $6.32488$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.133i·2-s + 15.9·4-s − 26.0i·5-s − 87.9i·7-s + 4.26i·8-s + 3.47·10-s + 101.·11-s + 190.·13-s + 11.7·14-s + 255.·16-s + 227.·17-s + 304. i·19-s − 416. i·20-s + 13.5i·22-s − 797.·23-s + ⋯
L(s)  = 1  + 0.0333i·2-s + 0.998·4-s − 1.04i·5-s − 1.79i·7-s + 0.0666i·8-s + 0.0347·10-s + 0.840·11-s + 1.12·13-s + 0.0598·14-s + 0.996·16-s + 0.788·17-s + 0.844i·19-s − 1.04i·20-s + 0.0280i·22-s − 1.50·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0745 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.0745 + 0.997i$
Analytic conductor: \(40.0041\)
Root analytic conductor: \(6.32488\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :2),\ -0.0745 + 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.931115520\)
\(L(\frac12)\) \(\approx\) \(2.931115520\)
\(L(3)\) 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 \( 1 + (137. - 1.84e3i)T \)
good2 \( 1 - 0.133iT - 16T^{2} \)
5 \( 1 + 26.0iT - 625T^{2} \)
7 \( 1 + 87.9iT - 2.40e3T^{2} \)
11 \( 1 - 101.T + 1.46e4T^{2} \)
13 \( 1 - 190.T + 2.85e4T^{2} \)
17 \( 1 - 227.T + 8.35e4T^{2} \)
19 \( 1 - 304. iT - 1.30e5T^{2} \)
23 \( 1 + 797.T + 2.79e5T^{2} \)
29 \( 1 + 1.08e3iT - 7.07e5T^{2} \)
31 \( 1 - 433.T + 9.23e5T^{2} \)
37 \( 1 + 310. iT - 1.87e6T^{2} \)
41 \( 1 + 1.33e3T + 2.82e6T^{2} \)
47 \( 1 + 1.00e3T + 4.87e6T^{2} \)
53 \( 1 - 3.64e3T + 7.89e6T^{2} \)
59 \( 1 - 1.28e3T + 1.21e7T^{2} \)
61 \( 1 + 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.75e3T + 2.01e7T^{2} \)
71 \( 1 - 4.25e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.32e3iT - 2.83e7T^{2} \)
79 \( 1 + 227.T + 3.89e7T^{2} \)
83 \( 1 - 1.89e3T + 4.74e7T^{2} \)
89 \( 1 + 1.08e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.09e4T + 8.85e7T^{2} \)
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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

−10.38711848623621788584134459705, −9.820581071347157253293925064225, −8.339521788569563801098201585152, −7.73807788477821744556708772019, −6.63609711365269333201227730280, −5.82429362426594130284986780310, −4.26288138818370758296475855346, −3.58589989922746669525333338538, −1.56767362324568181784524254789, −0.868949316201025504178161130801, 1.62943602244390280631334177963, 2.68779821358731099129893254950, 3.52964614336079915446348072828, 5.49293619627455962688167576940, 6.26608419390487242810077347357, 6.91447707032664409279334426271, 8.207352361726569770969118981981, 9.048783072452221560405749616699, 10.21487454818831699686167194815, 11.04045653820848990857003821988

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