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Properties

Label 2-1386-7.4-c1-0-4
Degree $2$
Conductor $1386$
Sign $0.386 - 0.922i$
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 + (1.5 + 2.59i)5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)10-s + (−0.5 + 0.866i)11-s + 6·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + (−3 − 5.19i)19-s − 3·20-s + 0.999·22-s + (2.5 + 4.33i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (0.474 − 0.821i)10-s + (−0.150 + 0.261i)11-s + 1.66·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + (−0.688 − 1.19i)19-s − 0.670·20-s + 0.213·22-s + (0.521 + 0.902i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.386 - 0.922i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151961829\)
\(L(\frac12)\) \(\approx\) \(1.151961829\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−9.916448842416488032827605078484, −9.006540446660709793182759469251, −8.400465084518878302233356933777, −7.04567856431553729244955602845, −6.59974913418605988984225055719, −5.83904825584983090881244272133, −4.34444960800290632302334056414, −3.36813368810486986421311984657, −2.69185802579767913790224877037, −1.42132518475133627083628390210, 0.55117606100553672993819145193, 1.86847297193809549682902764891, 3.35859804060016660428486740052, 4.54459492801154705183739351648, 5.45260091574327106129224170568, 6.20472428374815834006172146548, 6.70882626152288571719361777289, 8.105581476798569623220069197009, 8.747279706237932014664809760781, 9.118206553602416498153281073933

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