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Properties

Label 2-1386-7.2-c1-0-11
Degree $2$
Conductor $1386$
Sign $0.605 - 0.795i$
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 + (−2 − 1.73i)7-s + 0.999·8-s + (0.5 + 0.866i)11-s − 4·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s − 0.999·22-s + (−1.5 + 2.59i)23-s + (2.5 + 4.33i)25-s + (2 − 3.46i)26-s + (−0.5 + 2.59i)28-s + 9·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.150 + 0.261i)11-s − 1.10·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s − 0.213·22-s + (−0.312 + 0.541i)23-s + (0.5 + 0.866i)25-s + (0.392 − 0.679i)26-s + (−0.0944 + 0.490i)28-s + 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.100354199\)
\(L(\frac12)\) \(\approx\) \(1.100354199\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.585008766208023078537445409950, −9.032450223150904502466295136007, −7.81468494840802803262887427405, −7.36183957453432619304249379309, −6.53821597116820243167318879751, −5.71654506427053333093812376522, −4.68973275864814302578313536073, −3.78214763964455677313804430139, −2.52303483453212598594153913163, −0.901863701225170832255194409845, 0.70268164688304263305963699137, 2.44406063584878451689783852979, 2.96060924653939199280351827165, 4.24670786852414636716958178473, 5.17090620876012841715518721636, 6.24288980124215225267153106198, 7.03823068505168467452240858567, 8.068247593891087323826028353156, 8.762017923723495760083096159468, 9.652948014956762117777238651995

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