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Properties

Label 2-1386-77.10-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.571 - 0.820i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (2.01 − 1.16i)5-s + (1.31 + 2.29i)7-s + 0.999i·8-s + (−1.16 + 2.01i)10-s + (−2.46 − 2.21i)11-s + 1.44·13-s + (−2.28 − 1.32i)14-s + (−0.5 − 0.866i)16-s + (−1.60 + 2.78i)17-s + (3.07 + 5.33i)19-s − 2.32i·20-s + (3.24 + 0.686i)22-s + (3.14 + 5.44i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.901 − 0.520i)5-s + (0.497 + 0.867i)7-s + 0.353i·8-s + (−0.367 + 0.637i)10-s + (−0.743 − 0.668i)11-s + 0.399·13-s + (−0.611 − 0.355i)14-s + (−0.125 − 0.216i)16-s + (−0.389 + 0.675i)17-s + (0.706 + 1.22i)19-s − 0.520i·20-s + (0.691 + 0.146i)22-s + (0.656 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486643991\)
\(L(\frac12)\) \(\approx\) \(1.486643991\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.31 - 2.29i)T \)
11 \( 1 + (2.46 + 2.21i)T \)
good5 \( 1 + (-2.01 + 1.16i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 + (1.60 - 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.07 - 5.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.26iT - 29T^{2} \)
31 \( 1 + (-2.67 - 1.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.57 + 7.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 + 1.89iT - 43T^{2} \)
47 \( 1 + (-9.22 + 5.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.60 - 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.461 - 0.266i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.233 - 0.404i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 7.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 + (-2.50 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.21 + 4.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.56T + 83T^{2} \)
89 \( 1 + (-3.79 + 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−9.356986844652820557351146159949, −9.005175148024080266258928085392, −8.187042537585546451953414790515, −7.50379356045023123912488951956, −6.21560710860107069599560998953, −5.57695475655945481362183202820, −5.15476345007145618450853999700, −3.53260582619245824488818587975, −2.19776369605272011480965393061, −1.29264104463624033312282157053, 0.824609068669201868346207585975, 2.21370791766884413360532021878, 2.92600982220889248782891649927, 4.37800391766140479452750323890, 5.14878739444611571436560518910, 6.47674340113127959514740655853, 7.04473417330946972364647427493, 7.85093367969302985812876206863, 8.738527746716663239045216066121, 9.702558485563687860631706795432

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