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Properties

Label 2-1386-21.20-c1-0-20
Degree $2$
Conductor $1386$
Sign $0.401 + 0.915i$
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  − i·2-s − 4-s + 4.23·5-s + (−2.59 − 0.531i)7-s + i·8-s − 4.23i·10-s + i·11-s + (−0.531 + 2.59i)14-s + 16-s + 5.29·17-s + 0.250i·19-s − 4.23·20-s + 22-s − 4.50i·23-s + 12.9·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.89·5-s + (−0.979 − 0.200i)7-s + 0.353i·8-s − 1.33i·10-s + 0.301i·11-s + (−0.142 + 0.692i)14-s + 0.250·16-s + 1.28·17-s + 0.0573i·19-s − 0.946·20-s + 0.213·22-s − 0.938i·23-s + 2.58·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.401 + 0.915i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.091462082\)
\(L(\frac12)\) \(\approx\) \(2.091462082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (2.59 + 0.531i)T \)
11 \( 1 - iT \)
good5 \( 1 - 4.23T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 0.250iT - 19T^{2} \)
23 \( 1 + 4.50iT - 23T^{2} \)
29 \( 1 + 1.05iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 - 2.50T + 37T^{2} \)
41 \( 1 - 1.31T + 41T^{2} \)
43 \( 1 - 7.18T + 43T^{2} \)
47 \( 1 + 7.15T + 47T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 - 8.46T + 59T^{2} \)
61 \( 1 - 14.8iT - 61T^{2} \)
67 \( 1 - 9.42T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + 7.15iT - 73T^{2} \)
79 \( 1 + 0.377T + 79T^{2} \)
83 \( 1 + 1.84T + 83T^{2} \)
89 \( 1 - 1.56T + 89T^{2} \)
97 \( 1 + 7.96iT - 97T^{2} \)
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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

−9.497075496856794998251407303347, −9.162037754956023690500602069916, −7.916314232638488178828554965908, −6.75298409098336270136388174703, −5.99746500995490685219445763020, −5.38331094806921534736481012828, −4.20001095429789327614610504250, −2.96753851807256418847977134577, −2.25660079603352846919244387077, −1.02024913695357798336156823353, 1.25383369375988815557017479881, 2.63295065933807554501881683534, 3.59780700368762079436381445711, 5.24830369336892561715923907928, 5.56884082867383836016319027375, 6.41225480636102043578866794765, 6.97741256178110394000011464539, 8.171300495581820835794713922792, 9.080405228905874425120651189162, 9.734873860828158233947644518391

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