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Properties

Label 2-1386-77.10-c1-0-34
Degree $2$
Conductor $1386$
Sign $-0.190 + 0.981i$
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 − 1.16i)5-s + (−1.31 − 2.29i)7-s − 0.999i·8-s + (1.16 − 2.01i)10-s + (3.15 + 1.02i)11-s − 1.44·13-s + (−2.28 − 1.32i)14-s + (−0.5 − 0.866i)16-s + (1.60 − 2.78i)17-s + (−3.07 − 5.33i)19-s − 2.32i·20-s + (3.24 − 0.686i)22-s + (3.14 + 5.44i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.901 − 0.520i)5-s + (−0.497 − 0.867i)7-s − 0.353i·8-s + (0.367 − 0.637i)10-s + (0.950 + 0.309i)11-s − 0.399·13-s + (−0.611 − 0.355i)14-s + (−0.125 − 0.216i)16-s + (0.389 − 0.675i)17-s + (−0.706 − 1.22i)19-s − 0.520i·20-s + (0.691 − 0.146i)22-s + (0.656 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.190 + 0.981i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.640675213\)
\(L(\frac12)\) \(\approx\) \(2.640675213\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 2.29i)T \)
11 \( 1 + (-3.15 - 1.02i)T \)
good5 \( 1 + (-2.01 + 1.16i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + (-1.60 + 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.07 + 5.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.26iT - 29T^{2} \)
31 \( 1 + (-2.67 - 1.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.57 + 7.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 1.89iT - 43T^{2} \)
47 \( 1 + (-9.22 + 5.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.60 - 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.461 - 0.266i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.233 + 0.404i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 7.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 + (2.50 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.21 - 4.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.56T + 83T^{2} \)
89 \( 1 + (-3.79 + 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−9.443993890474398199508060892444, −8.934589896200475123253546501427, −7.39345549321404930155057186439, −6.86367506957086344491387369796, −5.92615202817467917963357315018, −5.04965509024953086278861812719, −4.25953009978473081691933511517, −3.25909609889091106464180406236, −2.06576787433567004861954188671, −0.893684294400661916132809506480, 1.79888329808114765969014065778, 2.80915988562235397183271550993, 3.72301598801631549791877287870, 4.88537159856858625746557415473, 5.97059136315017907924507829390, 6.24846716851202287558015841169, 7.01456501174916166090316091521, 8.320726006364192184853649201663, 8.850006413792256850912735648608, 9.925309767042192401582806158026

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