Properties

Label 2-1386-7.2-c1-0-24
Degree $2$
Conductor $1386$
Sign $0.701 + 0.712i$
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 − 1.73i)5-s + (2 − 1.73i)7-s + 0.999·8-s + (0.999 + 1.73i)10-s + (−0.5 − 0.866i)11-s + 2·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1.5 − 2.59i)19-s − 1.99·20-s + 0.999·22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + (0.755 − 0.654i)7-s + 0.353·8-s + (0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + 0.554·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + (0.344 − 0.596i)19-s − 0.447·20-s + 0.213·22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.701 + 0.712i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.539303257\)
\(L(\frac12)\) \(\approx\) \(1.539303257\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.311935652411048986490247624879, −8.597306716603120223905779381596, −7.970832409439989107870165291578, −7.09309338290594131659067586554, −6.24827220697949292882323586827, −5.17793386155280639859768629554, −4.76409147569165576349404978167, −3.50768630341792236030428700600, −1.83489091306361487411222903180, −0.76425809182926482829844308124, 1.47521264060918931232613496761, 2.40889638730991916345205184992, 3.34972173498370529359168213142, 4.51173447792363262900651372980, 5.53491908217530440620002485895, 6.38491491524175610387571592154, 7.38597638897484079951488206015, 8.256929195971284022731157773286, 8.863239420121568335880281049672, 9.889477312920301324964904860507

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