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Properties

Label 2-1386-7.2-c1-0-15
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 + 3.46i)5-s + (−2.5 + 0.866i)7-s + 0.999·8-s + (−1.99 − 3.46i)10-s + (−0.5 − 0.866i)11-s − 13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (−3 + 5.19i)19-s + 3.99·20-s + 0.999·22-s + (−1 + 1.73i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.894 + 1.54i)5-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (−0.632 − 1.09i)10-s + (−0.150 − 0.261i)11-s − 0.277·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.688 + 1.19i)19-s + 0.894·20-s + 0.213·22-s + (−0.208 + 0.361i)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: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
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 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.5 + 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−9.502920509103336239377245719260, −8.381601595198172497295150491774, −7.72911967490789593129587584884, −7.01367227274302486257420815689, −6.27751924547485755292347136030, −5.66563171531911962519381403483, −4.05122305797424573947698127282, −3.41286312951143572308018911181, −2.29754830802319300278492286719, 0, 1.00503494855272045806181227706, 2.57653131589788772638924663908, 3.75101847876993285462280275100, 4.49395480179660569319583333273, 5.25034017071322233503876085325, 6.62760607304414910483891663431, 7.54603717305674134171837986224, 8.237682208326296493849964574679, 9.201647939686874887845029844992

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