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Properties

Label 2-1386-77.54-c1-0-1
Degree $2$
Conductor $1386$
Sign $-0.798 - 0.602i$
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 + (−1.23 − 0.712i)5-s + (−0.484 + 2.60i)7-s − 0.999i·8-s + (0.712 + 1.23i)10-s + (2.51 − 2.16i)11-s − 2.39·13-s + (1.72 − 2.01i)14-s + (−0.5 + 0.866i)16-s + (0.529 + 0.917i)17-s + (−2.37 + 4.11i)19-s − 1.42i·20-s + (−3.25 + 0.620i)22-s + (1.72 − 2.97i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.551 − 0.318i)5-s + (−0.183 + 0.983i)7-s − 0.353i·8-s + (0.225 + 0.390i)10-s + (0.757 − 0.653i)11-s − 0.663·13-s + (0.459 − 0.537i)14-s + (−0.125 + 0.216i)16-s + (0.128 + 0.222i)17-s + (−0.544 + 0.943i)19-s − 0.318i·20-s + (−0.694 + 0.132i)22-s + (0.358 − 0.621i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.798 - 0.602i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.798 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2095511352\)
\(L(\frac12)\) \(\approx\) \(0.2095511352\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.484 - 2.60i)T \)
11 \( 1 + (-2.51 + 2.16i)T \)
good5 \( 1 + (1.23 + 0.712i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.39T + 13T^{2} \)
17 \( 1 + (-0.529 - 0.917i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.72 + 2.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.13iT - 29T^{2} \)
31 \( 1 + (-0.122 + 0.0706i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.19 + 2.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.17T + 41T^{2} \)
43 \( 1 - 5.73iT - 43T^{2} \)
47 \( 1 + (3.08 + 1.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.98 + 3.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.71 - 2.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.58 - 9.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.567 + 0.983i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (0.969 + 1.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.82 + 2.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.73T + 83T^{2} \)
89 \( 1 + (1.67 + 0.969i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.9iT - 97T^{2} \)
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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

−9.836758954577171678991187232680, −8.931817657513375082783727208658, −8.505766859452751692366364025483, −7.74764010317983709888973328196, −6.64568744041810327904513699615, −5.91014285393189794631366218805, −4.75631835023363877515207986857, −3.71557262907265432510650610594, −2.75613396698320893848759285937, −1.52875046096884729730563995926, 0.10558458854054646877621185023, 1.62116698295544996977875488659, 3.08098162042111562638393194637, 4.15999933977453608142890038572, 4.97121707674539076726812663206, 6.29421251104569116680378458019, 7.11693572802325511694762293041, 7.39804254368982699116375704315, 8.371963972212830157426161234001, 9.399618708234088595322391897251

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