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Properties

Label 2-1386-77.54-c1-0-2
Degree $2$
Conductor $1386$
Sign $-0.794 - 0.607i$
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 + (2.53 + 1.46i)5-s + (−2.00 − 1.72i)7-s − 0.999i·8-s + (−1.46 − 2.53i)10-s + (−3.12 − 1.10i)11-s − 1.09·13-s + (0.874 + 2.49i)14-s + (−0.5 + 0.866i)16-s + (−2.00 − 3.47i)17-s + (−3.32 + 5.76i)19-s + 2.92i·20-s + (2.15 + 2.51i)22-s + (0.874 − 1.51i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.13 + 0.653i)5-s + (−0.758 − 0.652i)7-s − 0.353i·8-s + (−0.461 − 0.800i)10-s + (−0.943 − 0.332i)11-s − 0.302·13-s + (0.233 + 0.667i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.841i)17-s + (−0.763 + 1.32i)19-s + 0.653i·20-s + (0.459 + 0.537i)22-s + (0.182 − 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.794 - 0.607i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2564182405\)
\(L(\frac12)\) \(\approx\) \(0.2564182405\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 + 1.72i)T \)
11 \( 1 + (3.12 + 1.10i)T \)
good5 \( 1 + (-2.53 - 1.46i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 + (2.00 + 3.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.32 - 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.874 + 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.40iT - 29T^{2} \)
31 \( 1 + (7.56 - 4.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.36 - 9.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.63T + 41T^{2} \)
43 \( 1 - 1.44iT - 43T^{2} \)
47 \( 1 + (1.67 + 0.969i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.32 + 5.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.85 + 3.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.37 - 2.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.70 + 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + (4.01 + 6.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.41 + 4.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 + (6.94 + 4.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−10.06926624551825931448711419336, −9.277172170324757254385497041399, −8.436495320022721170325977455609, −7.37956649326812992395799892168, −6.74442443704319863887884210946, −5.96972514756341833435934107835, −4.92237380894393031995181661489, −3.48123228655660226324206338045, −2.74577891627835104422145606874, −1.67652046258144930937879448677, 0.11446219008206374838201564358, 1.95041833428103909381910839551, 2.58732543719382011353552512788, 4.29751535232536060390159533029, 5.47780845480320787910803763638, 5.82197457071927006009844937881, 6.79849337148298995442644152595, 7.66354739126525198752577964010, 8.800987021655444640952932431155, 9.121703885642746254728926857689

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