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Properties

Label 2-1386-33.8-c1-0-17
Degree $2$
Conductor $1386$
Sign $-0.559 + 0.828i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (1.25 − 1.72i)5-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + 2.12i·10-s + (−2.72 + 1.88i)11-s + (0.138 + 0.190i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + (−3.83 − 2.78i)17-s + (2.37 − 0.770i)19-s + (−1.25 − 1.72i)20-s + (1.09 − 3.13i)22-s − 7.44i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.559 − 0.770i)5-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s + 0.673i·10-s + (−0.821 + 0.569i)11-s + (0.0383 + 0.0527i)13-s + (0.254 − 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−0.931 − 0.676i)17-s + (0.543 − 0.176i)19-s + (−0.279 − 0.385i)20-s + (0.233 − 0.667i)22-s − 1.55i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.559 + 0.828i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.559 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5927265762\)
\(L(\frac12)\) \(\approx\) \(0.5927265762\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (2.72 - 1.88i)T \)
good5 \( 1 + (-1.25 + 1.72i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.138 - 0.190i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.83 + 2.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.37 + 0.770i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.44iT - 23T^{2} \)
29 \( 1 + (1.54 - 4.75i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.902 + 0.656i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.09 + 9.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.856 + 2.63i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.37iT - 43T^{2} \)
47 \( 1 + (11.7 - 3.80i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.73 + 6.51i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (12.9 + 4.20i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.05 - 9.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.74T + 67T^{2} \)
71 \( 1 + (5.46 - 7.51i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.11 - 1.33i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.17 + 5.74i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.89 + 1.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.21iT - 89T^{2} \)
97 \( 1 + (-0.932 + 0.677i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.176077376610361234800623165153, −8.704395134883452856725100999727, −7.64786633991276249649984330770, −6.96322720186151164905646033413, −6.06263166181546882445823479481, −5.12710531676783399851574955395, −4.51120765154828893325041308759, −2.87269668966942112415148718725, −1.77573204163839835034379964906, −0.27717503810350769515153647982, 1.61476792131437734462278014446, 2.76812025435370666802065577016, 3.40614674177231478166488979330, 4.78384264580872360932811967400, 6.01457061270780014957380505715, 6.48433324270928429302128978220, 7.64814964821084258344890814777, 8.202497765595989207179254719730, 9.288025932848763297058682307882, 9.853525608666544409933093183845

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