Properties

Label 2-1386-77.76-c1-0-1
Degree $2$
Conductor $1386$
Sign $-0.818 + 0.575i$
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  + i·2-s − 4-s + 0.646i·5-s + (2.44 + i)7-s i·8-s − 0.646·10-s + (−2.79 − 1.79i)11-s − 3.09·13-s + (−1 + 2.44i)14-s + 16-s − 3.74·17-s − 5.54·19-s − 0.646i·20-s + (1.79 − 2.79i)22-s − 4·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.288i·5-s + (0.925 + 0.377i)7-s − 0.353i·8-s − 0.204·10-s + (−0.841 − 0.540i)11-s − 0.858·13-s + (−0.267 + 0.654i)14-s + 0.250·16-s − 0.907·17-s − 1.27·19-s − 0.144i·20-s + (0.381 − 0.595i)22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.818 + 0.575i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.818 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3484802064\)
\(L(\frac12)\) \(\approx\) \(0.3484802064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.44 - i)T \)
11 \( 1 + (2.79 + 1.79i)T \)
good5 \( 1 - 0.646iT - 5T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.58iT - 29T^{2} \)
31 \( 1 + 1.15iT - 31T^{2} \)
37 \( 1 + 5.58T + 37T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 + 5.03iT - 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 + 3.09iT - 59T^{2} \)
61 \( 1 + 9.28T + 61T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 + 15.9iT - 97T^{2} \)
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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

−10.10975072242941527611152781429, −8.844625311322821311704352027082, −8.500714154532331950852086135198, −7.62480072870278002279339497565, −6.84472729598756716024855387930, −5.96690444497082575979600804959, −5.03468510411286841875783342983, −4.46816497701266335204399505408, −3.03684669497185071045077912079, −1.92430273093018105522092504308, 0.13063235393916447917351413837, 1.83718892309898971538699177901, 2.53761138395734820327141860571, 4.10009880352403347777561178965, 4.65228969717493853598518622416, 5.43889134487594145638606215481, 6.73993079734012989180684356485, 7.65544020786271130556750879237, 8.388997575106378603937073990968, 9.078198593159204957625293071808

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