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Properties

Label 2-1386-7.4-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.0725 + 0.997i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.207 + 0.358i)5-s + (−2.62 + 0.358i)7-s + 0.999·8-s + (0.207 − 0.358i)10-s + (−0.5 + 0.866i)11-s − 1.17·13-s + (1.62 + 2.09i)14-s + (−0.5 − 0.866i)16-s + (1.08 − 1.88i)17-s + (−0.414 − 0.717i)19-s − 0.414·20-s + 0.999·22-s + (1.62 + 2.80i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0926 + 0.160i)5-s + (−0.990 + 0.135i)7-s + 0.353·8-s + (0.0654 − 0.113i)10-s + (−0.150 + 0.261i)11-s − 0.324·13-s + (0.433 + 0.558i)14-s + (−0.125 − 0.216i)16-s + (0.263 − 0.456i)17-s + (−0.0950 − 0.164i)19-s − 0.0926·20-s + 0.213·22-s + (0.338 + 0.585i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9902969648\)
\(L(\frac12)\) \(\approx\) \(0.9902969648\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.358i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.207 - 0.358i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + (-1.08 + 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.414 + 0.717i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.62 - 2.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-3.24 + 5.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.65T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 + (-4.62 - 8.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.58 + 4.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.82 + 3.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.792 - 1.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.74 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-2.41 + 4.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.37 + 4.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.82T + 83T^{2} \)
89 \( 1 + (6.24 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.529818951296537360587761561041, −8.825608951632944417830845977678, −7.80542594501443287958165685185, −7.02897476909572800431686160951, −6.16901182453900406748280458673, −5.11056901136977038405331426200, −4.04172649309179217644802879903, −3.02829206909068152429715690932, −2.24354150718876193339065855400, −0.55407289796744983334991458105, 1.04621127453767368454729911395, 2.70190915678549223913815716721, 3.75773428502498641542591272061, 4.92718593687242269197684456522, 5.73394796241594168143685231731, 6.68078713577740265998364539213, 7.13298533054877379104933178456, 8.340372455464620669009341998868, 8.774630435469290771339669069384, 9.819700189522624510861094223997

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