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Properties

Label 2-1386-77.10-c1-0-4
Degree $2$
Conductor $1386$
Sign $-0.0387 - 0.999i$
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 + (−0.194 + 0.112i)5-s + (−2.62 − 0.307i)7-s − 0.999i·8-s + (−0.112 + 0.194i)10-s + (1.73 + 2.82i)11-s − 5.03·13-s + (−2.42 + 1.04i)14-s + (−0.5 − 0.866i)16-s + (−3.92 + 6.80i)17-s + (3.66 + 6.34i)19-s + 0.224i·20-s + (2.91 + 1.57i)22-s + (−2.42 − 4.20i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0868 + 0.0501i)5-s + (−0.993 − 0.116i)7-s − 0.353i·8-s + (−0.0354 + 0.0613i)10-s + (0.524 + 0.851i)11-s − 1.39·13-s + (−0.649 + 0.280i)14-s + (−0.125 − 0.216i)16-s + (−0.952 + 1.65i)17-s + (0.840 + 1.45i)19-s + 0.0501i·20-s + (0.622 + 0.336i)22-s + (−0.506 − 0.877i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0387 - 0.999i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.0387 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.096419742\)
\(L(\frac12)\) \(\approx\) \(1.096419742\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.307i)T \)
11 \( 1 + (-1.73 - 2.82i)T \)
good5 \( 1 + (0.194 - 0.112i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.03T + 13T^{2} \)
17 \( 1 + (3.92 - 6.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.66 - 6.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.42 + 4.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (-3.73 - 2.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0891 + 0.154i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.568T + 41T^{2} \)
43 \( 1 - 6.39iT - 43T^{2} \)
47 \( 1 + (-2.93 + 1.69i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.07 - 1.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.00 + 1.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.32 + 7.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (5.25 - 9.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.75 - 5.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (-9.10 + 5.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.96iT - 97T^{2} \)
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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

−10.03786916881977279188039121107, −9.256051004581615079335160371930, −8.066547929547155253423691587932, −7.18927316662320656202103546044, −6.38523654205736808990103791027, −5.70348146698977551623505930349, −4.42299577998539272796009115771, −3.90448025976015558058548603183, −2.73705544029442168139856112818, −1.69409570400247811721007197659, 0.33852319638525349467473920963, 2.53488392172183265105432117935, 3.16048741778267072228807023232, 4.36324342086206914394174828767, 5.16433169237973120866520695294, 6.04656136251040932725592389328, 7.05536463172712636005636272250, 7.29009357219494980746299162883, 8.665397569823076572050669660693, 9.336362788573561915437318236986

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