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Properties

Label 2-1386-77.76-c1-0-8
Degree $2$
Conductor $1386$
Sign $-0.320 - 0.947i$
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 + 4.14i·5-s + (0.717 + 2.54i)7-s + i·8-s + 4.14·10-s + (0.170 + 3.31i)11-s + 3.09·13-s + (2.54 − 0.717i)14-s + 16-s − 3.57·17-s + 3.91·19-s − 4.14i·20-s + (3.31 − 0.170i)22-s − 7.71·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.85i·5-s + (0.271 + 0.962i)7-s + 0.353i·8-s + 1.30·10-s + (0.0514 + 0.998i)11-s + 0.857·13-s + (0.680 − 0.191i)14-s + 0.250·16-s − 0.867·17-s + 0.898·19-s − 0.926i·20-s + (0.706 − 0.0363i)22-s − 1.60·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287304840\)
\(L(\frac12)\) \(\approx\) \(1.287304840\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.717 - 2.54i)T \)
11 \( 1 + (-0.170 - 3.31i)T \)
good5 \( 1 - 4.14iT - 5T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 + 7.71T + 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 + 9.23iT - 31T^{2} \)
37 \( 1 + 0.869T + 37T^{2} \)
41 \( 1 - 5.91T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 6.76iT - 47T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 + 1.33T + 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 + 4.52iT - 97T^{2} \)
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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

−9.780058851698506948921288611249, −9.426062477331955871675963558962, −8.107422036254084130149149963817, −7.52645759089489027512809271271, −6.35084252582815249452863037730, −5.91397326695875366921114799410, −4.50051943665995487586773953688, −3.58817882032924035483537060147, −2.56062406073731566076153448520, −1.97543199715181090156751328727, 0.54013348543344178102291519870, 1.50977011840338733263347404974, 3.67526288507687129292379859990, 4.29274835760470777004631625405, 5.24231102134730124541655376776, 5.84743285532436234602780292510, 6.92957368764809320441587941250, 7.903712008654293106576352274841, 8.577910101052140245005834540235, 8.919333176005802861861684588207

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