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Properties

Label 2-1386-33.32-c1-0-8
Degree $2$
Conductor $1386$
Sign $0.527 - 0.849i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + 2-s + 4-s + 0.494i·5-s − i·7-s + 8-s + 0.494i·10-s + (0.197 + 3.31i)11-s + 5.50i·13-s − i·14-s + 16-s − 2.35·17-s + 1.14i·19-s + 0.494i·20-s + (0.197 + 3.31i)22-s + 4.19i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.220i·5-s − 0.377i·7-s + 0.353·8-s + 0.156i·10-s + (0.0596 + 0.998i)11-s + 1.52i·13-s − 0.267i·14-s + 0.250·16-s − 0.571·17-s + 0.263i·19-s + 0.110i·20-s + (0.0421 + 0.705i)22-s + 0.874i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.527 - 0.849i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.489175217\)
\(L(\frac12)\) \(\approx\) \(2.489175217\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + iT \)
11 \( 1 + (-0.197 - 3.31i)T \)
good5 \( 1 - 0.494iT - 5T^{2} \)
13 \( 1 - 5.50iT - 13T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 - 1.14iT - 19T^{2} \)
23 \( 1 - 4.19iT - 23T^{2} \)
29 \( 1 + 0.175T + 29T^{2} \)
31 \( 1 - 1.40T + 31T^{2} \)
37 \( 1 - 4.71T + 37T^{2} \)
41 \( 1 - 3.80T + 41T^{2} \)
43 \( 1 + 9.72iT - 43T^{2} \)
47 \( 1 - 4.06iT - 47T^{2} \)
53 \( 1 - 2.97iT - 53T^{2} \)
59 \( 1 - 2.26iT - 59T^{2} \)
61 \( 1 + 4.51iT - 61T^{2} \)
67 \( 1 - 8.46T + 67T^{2} \)
71 \( 1 - 3.20iT - 71T^{2} \)
73 \( 1 - 2.39iT - 73T^{2} \)
79 \( 1 - 1.55iT - 79T^{2} \)
83 \( 1 + 4.29T + 83T^{2} \)
89 \( 1 - 2.32iT - 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−9.691128210580823176417269141886, −9.055214459904119554957943247034, −7.87138197335919359896911101402, −6.97689471413389446140693568838, −6.62019831338180700248529053233, −5.44205784155270781389444872935, −4.45325186430948953768322733827, −3.93424405500694338498220124068, −2.60408444615964044810854952154, −1.59078064153918742971068898342, 0.830385289769384061150848008058, 2.54832265283199589329117170730, 3.23421764756648507931111547028, 4.42447450525870494062424536254, 5.27539534158981071445692271915, 6.01122900172401749009854099957, 6.76770815083007903137042662239, 7.955883367112365531360906575728, 8.470721484817596939707444997475, 9.412921882298905109742734103769

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