Properties

Degree $2$
Conductor $1386$
Sign $-0.962 + 0.271i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s − 1.06·5-s + (0.884 − 2.49i)7-s + i·8-s + 1.06i·10-s + i·11-s + (−2.49 − 0.884i)14-s + 16-s + 3.91·17-s − 7.49i·19-s + 1.06·20-s + 22-s + 5.33i·23-s − 3.85·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.478·5-s + (0.334 − 0.942i)7-s + 0.353i·8-s + 0.338i·10-s + 0.301i·11-s + (−0.666 − 0.236i)14-s + 0.250·16-s + 0.950·17-s − 1.71i·19-s + 0.239·20-s + 0.213·22-s + 1.11i·23-s − 0.771·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.962 + 0.271i$
Motivic weight: \(1\)
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.962 + 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.012115018\)
\(L(\frac12)\) \(\approx\) \(1.012115018\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.884 + 2.49i)T \)
11 \( 1 - iT \)
good5 \( 1 + 1.06T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3.91T + 17T^{2} \)
19 \( 1 + 7.49iT - 19T^{2} \)
23 \( 1 - 5.33iT - 23T^{2} \)
29 \( 1 + 8.01iT - 29T^{2} \)
31 \( 1 + 3.65iT - 31T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 0.230T + 43T^{2} \)
47 \( 1 + 0.370T + 47T^{2} \)
53 \( 1 - 7.56iT - 53T^{2} \)
59 \( 1 + 2.13T + 59T^{2} \)
61 \( 1 + 14.2iT - 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 + 3.15iT - 71T^{2} \)
73 \( 1 + 0.370iT - 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−9.516916193016445405175050284711, −8.364530258483650138777188317111, −7.64284258529484634471979375562, −7.00700342183080894423327494577, −5.70554757453599763909110716465, −4.70641723935617823793519562758, −3.98798202755529801348075037899, −3.08339268355336790263202978940, −1.75314653620608062226788221596, −0.42641383156806202489021768208, 1.56256071293657096943907100737, 3.10776495201116686686971568355, 4.01524361781203817235725187035, 5.20929474987850721043754999883, 5.71517554574045581962260309450, 6.67822853836865594931003511768, 7.61362598379776062083586054714, 8.383760920162532913147051937068, 8.750681646546180773827771073261, 9.900755128330339047252679913021

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