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Properties

Label 2-1386-231.32-c1-0-5
Degree $2$
Conductor $1386$
Sign $-0.999 - 0.00620i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.42 + 0.822i)5-s + (2.58 + 0.551i)7-s − 0.999·8-s + (−1.42 − 0.822i)10-s + (−2.53 + 2.13i)11-s − 1.29i·13-s + (0.816 + 2.51i)14-s + (−0.5 − 0.866i)16-s + (−1.68 + 2.91i)17-s + (−0.340 + 0.196i)19-s − 1.64i·20-s + (−3.11 − 1.13i)22-s + (−0.455 + 0.262i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.636 + 0.367i)5-s + (0.978 + 0.208i)7-s − 0.353·8-s + (−0.450 − 0.259i)10-s + (−0.765 + 0.643i)11-s − 0.358i·13-s + (0.218 + 0.672i)14-s + (−0.125 − 0.216i)16-s + (−0.408 + 0.707i)17-s + (−0.0780 + 0.0450i)19-s − 0.367i·20-s + (−0.664 − 0.241i)22-s + (−0.0949 + 0.0548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.00620i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.999 - 0.00620i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.085161336\)
\(L(\frac12)\) \(\approx\) \(1.085161336\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.58 - 0.551i)T \)
11 \( 1 + (2.53 - 2.13i)T \)
good5 \( 1 + (1.42 - 0.822i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 1.29iT - 13T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.340 - 0.196i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.455 - 0.262i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 + (2.86 - 4.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.50 - 2.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + (4.48 - 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.43 + 0.830i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.26 - 4.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 1.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.62iT - 71T^{2} \)
73 \( 1 + (8.82 + 5.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.22 + 4.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.98T + 83T^{2} \)
89 \( 1 + (8.32 - 4.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−10.10706507381657623763739843870, −8.789267095097162992186885807586, −8.257328604019824557506980493886, −7.48641106619316350519634391190, −6.90745635401665458153574261642, −5.71904481798262293930975913615, −5.00190002814603723365262112054, −4.17366513591653598085127359680, −3.14463339555017094736165989168, −1.85625972181798028394017615989, 0.37910373953835252573317218889, 1.81929921468656685374242857100, 2.96893533026761409123857389362, 4.10863455732502725025003138149, 4.76540184806355481147077132959, 5.54314900215426722760212093079, 6.69369800352468488204905235350, 7.82181740623109077467497065362, 8.286837819899681476054658431048, 9.188636393757497566279847101981

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