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Properties

Label 2-1386-21.5-c1-0-17
Degree $2$
Conductor $1386$
Sign $0.999 - 0.0213i$
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 − 3.48i)5-s + (2.64 − 0.148i)7-s + 0.999i·8-s + (3.48 − 2.01i)10-s + (−0.866 + 0.5i)11-s + 5.78i·13-s + (2.36 + 1.19i)14-s + (−0.5 + 0.866i)16-s + (−0.655 − 1.13i)17-s + (5.95 + 3.43i)19-s + 4.02·20-s − 0.999·22-s + (2.08 + 1.20i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.900 − 1.55i)5-s + (0.998 − 0.0562i)7-s + 0.353i·8-s + (1.10 − 0.636i)10-s + (−0.261 + 0.150i)11-s + 1.60i·13-s + (0.631 + 0.318i)14-s + (−0.125 + 0.216i)16-s + (−0.159 − 0.275i)17-s + (1.36 + 0.788i)19-s + 0.900·20-s − 0.213·22-s + (0.434 + 0.250i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0213i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.999 - 0.0213i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.173673610\)
\(L(\frac12)\) \(\approx\) \(3.173673610\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.148i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-2.01 + 3.48i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.78iT - 13T^{2} \)
17 \( 1 + (0.655 + 1.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.95 - 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.08 - 1.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.530iT - 29T^{2} \)
31 \( 1 + (2.04 - 1.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 7.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 - 0.642T + 43T^{2} \)
47 \( 1 + (-5.99 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.35 + 1.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.834 + 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.2 + 7.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.09 + 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.08i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.70 - 9.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (-1.06 + 1.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−9.235831420963194681857091350112, −8.929019795403076330170943170644, −7.916889620359935591131171043615, −7.16417560389621389693409997287, −5.99786318042501781025568176446, −5.23035426137434598152956907869, −4.77306469812115359792105774104, −3.87288890773517749923069586487, −2.14164320846442870351126189094, −1.35966859844971504238869265624, 1.39911372952899215879613850112, 2.83937726451989015670303565656, 2.96826877456363196422491943175, 4.53693362976709439608922606546, 5.56826413451149475307006083957, 5.96535816631847861527428905652, 7.16571243518025951036168275887, 7.66050998598791738172532506656, 8.860379500133353629324803498425, 10.01294424071595044970299409641

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