Properties

Label 2-1386-77.76-c1-0-19
Degree $2$
Conductor $1386$
Sign $-0.195 - 0.980i$
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 + 3.06i·5-s + (2.37 − 1.16i)7-s i·8-s − 3.06·10-s + (3.20 + 0.857i)11-s + 0.338·13-s + (1.16 + 2.37i)14-s + 16-s + 0.314·17-s + 6.09·19-s − 3.06i·20-s + (−0.857 + 3.20i)22-s + 3.37·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.36i·5-s + (0.897 − 0.441i)7-s − 0.353i·8-s − 0.968·10-s + (0.966 + 0.258i)11-s + 0.0939·13-s + (0.312 + 0.634i)14-s + 0.250·16-s + 0.0762·17-s + 1.39·19-s − 0.684i·20-s + (−0.182 + 0.683i)22-s + 0.703·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.195 - 0.980i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934296645\)
\(L(\frac12)\) \(\approx\) \(1.934296645\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 + 1.16i)T \)
11 \( 1 + (-3.20 - 0.857i)T \)
good5 \( 1 - 3.06iT - 5T^{2} \)
13 \( 1 - 0.338T + 13T^{2} \)
17 \( 1 - 0.314T + 17T^{2} \)
19 \( 1 - 6.09T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + 4.40iT - 29T^{2} \)
31 \( 1 + 0.722iT - 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 - 4.12iT - 43T^{2} \)
47 \( 1 - 8.77iT - 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 - 4.67iT - 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 1.68T + 73T^{2} \)
79 \( 1 + 6.86iT - 79T^{2} \)
83 \( 1 + 7.11T + 83T^{2} \)
89 \( 1 + 2.28iT - 89T^{2} \)
97 \( 1 - 5.08iT - 97T^{2} \)
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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

−9.737710254567874838512287668313, −8.984239713899964996255940944654, −7.84546936869375304995963297598, −7.35999119384078522696139908020, −6.67524673683373235367194909541, −5.85260174774581762894976965786, −4.78827406232670172505739667369, −3.86609287095921048011132894518, −2.87578152711925302398085513028, −1.33240927364703271562904557979, 0.991253240575406569018903305665, 1.71735484921011989379228849115, 3.21180686640055451465132441381, 4.25130568702846784281418802628, 5.11677063325223633154307444009, 5.57665131075024776679866867305, 7.01212871892478192100508223710, 8.062800300718378117833225954892, 8.844988223163179686126066756050, 9.115270904133268249462074516469

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