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Properties

Label 2-1386-7.4-c1-0-23
Degree $2$
Conductor $1386$
Sign $0.0725 + 0.997i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.207 − 0.358i)5-s + (−2.62 + 0.358i)7-s − 0.999·8-s + (0.207 − 0.358i)10-s + (0.5 − 0.866i)11-s − 1.17·13-s + (−1.62 − 2.09i)14-s + (−0.5 − 0.866i)16-s + (−1.08 + 1.88i)17-s + (−0.414 − 0.717i)19-s + 0.414·20-s + 0.999·22-s + (−1.62 − 2.80i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0926 − 0.160i)5-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (0.0654 − 0.113i)10-s + (0.150 − 0.261i)11-s − 0.324·13-s + (−0.433 − 0.558i)14-s + (−0.125 − 0.216i)16-s + (−0.263 + 0.456i)17-s + (−0.0950 − 0.164i)19-s + 0.0926·20-s + 0.213·22-s + (−0.338 − 0.585i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6445626573\)
\(L(\frac12)\) \(\approx\) \(0.6445626573\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.358i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.207 + 0.358i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + (1.08 - 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.414 + 0.717i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 + 2.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + (-3.24 + 5.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 + (4.62 + 8.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.58 - 4.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.82 - 3.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.792 - 1.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.74 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (-2.41 + 4.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.37 + 4.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 + (-6.24 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.220357982060839557940533711501, −8.560369637468743659283574216933, −7.70742617316236299972913544581, −6.72077057347371523663387637081, −6.22618291882989683056977629327, −5.28979939127568708750041287234, −4.27276199434790416563689152513, −3.44163445302693279683894351679, −2.30229617999755516831390922854, −0.22659812491714184598963774322, 1.51528882943935289287348409276, 2.88633330558794700538444226758, 3.51848597231228586258581817514, 4.61817547246575413571789199545, 5.47412752164126205848966644415, 6.56569202647164996088199418031, 7.09764834662011395415197378905, 8.272343244147741913944752118207, 9.241628047966086435298490054665, 9.849666379413090841142916576333

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