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Properties

Label 2-1386-33.8-c1-0-21
Degree $2$
Conductor $1386$
Sign $-0.921 + 0.388i$
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 + (0.943 − 1.29i)5-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + 1.60i·10-s + (1.01 − 3.15i)11-s + (−1.22 − 1.68i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + (−4.11 − 2.98i)17-s + (−6.93 + 2.25i)19-s + (−0.943 − 1.29i)20-s + (1.03 + 3.15i)22-s + 0.571i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.422 − 0.580i)5-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s + 0.507i·10-s + (0.306 − 0.951i)11-s + (−0.338 − 0.466i)13-s + (0.254 − 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−0.997 − 0.724i)17-s + (−1.59 + 0.516i)19-s + (−0.211 − 0.290i)20-s + (0.220 + 0.671i)22-s + 0.119i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.921 + 0.388i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3279220532\)
\(L(\frac12)\) \(\approx\) \(0.3279220532\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.01 + 3.15i)T \)
good5 \( 1 + (-0.943 + 1.29i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (1.22 + 1.68i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.11 + 2.98i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.93 - 2.25i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.571iT - 23T^{2} \)
29 \( 1 + (-0.282 + 0.867i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.53 - 4.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0690 - 0.212i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.429 + 1.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.30iT - 43T^{2} \)
47 \( 1 + (-2.53 + 0.824i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.49 + 2.05i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.53 + 0.823i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.63 - 6.37i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 + (1.60 - 2.20i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.42 + 2.73i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.22 + 1.68i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.76 + 2.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.76iT - 89T^{2} \)
97 \( 1 + (-5.45 + 3.96i)T + (29.9 - 92.2i)T^{2} \)
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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

−8.984393706526426486263887945934, −8.684999676646633195759755071087, −7.66912852283779741230253371406, −6.74041962638164648550279504196, −6.03014974099974366611014788155, −5.21915492811562641202713439715, −4.18714492408738892729984016044, −2.88168006665629606290724652373, −1.58467436334207655502906440344, −0.15085834624703189589964654865, 1.93602270898845560952604393493, 2.47373804055963844400235819096, 3.88268249385314233064611806234, 4.64962000254181698670465165245, 6.14977906282576845299426006092, 6.70307064467642149028704433845, 7.43201482896196211324433044333, 8.611569139685919903585268035337, 9.143005396911413081794840982475, 9.966778380479916116446542031949

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