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Properties

Label 2-1386-33.29-c1-0-7
Degree $2$
Conductor $1386$
Sign $-0.575 - 0.817i$
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 + (1.04 + 1.43i)5-s + (−0.951 + 0.309i)7-s + (−0.309 + 0.951i)8-s + 1.77i·10-s + (0.808 − 3.21i)11-s + (−2.87 + 3.95i)13-s + (−0.951 − 0.309i)14-s + (−0.809 + 0.587i)16-s + (−3.29 + 2.39i)17-s + (6.36 + 2.06i)19-s + (−1.04 + 1.43i)20-s + (2.54 − 2.12i)22-s + 8.52i·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.466 + 0.641i)5-s + (−0.359 + 0.116i)7-s + (−0.109 + 0.336i)8-s + 0.561i·10-s + (0.243 − 0.969i)11-s + (−0.796 + 1.09i)13-s + (−0.254 − 0.0825i)14-s + (−0.202 + 0.146i)16-s + (−0.798 + 0.580i)17-s + (1.46 + 0.474i)19-s + (−0.233 + 0.320i)20-s + (0.542 − 0.453i)22-s + 1.77i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.575 - 0.817i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053699105\)
\(L(\frac12)\) \(\approx\) \(2.053699105\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.808 + 3.21i)T \)
good5 \( 1 + (-1.04 - 1.43i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.87 - 3.95i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.29 - 2.39i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.36 - 2.06i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.52iT - 23T^{2} \)
29 \( 1 + (-1.42 - 4.38i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.200 - 0.617i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.28 - 10.1i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.80iT - 43T^{2} \)
47 \( 1 + (-1.05 - 0.344i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.49 + 7.55i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.65 + 1.18i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.09 + 7.01i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.30T + 67T^{2} \)
71 \( 1 + (-0.602 - 0.829i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.69 - 1.20i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.18 + 12.6i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.47 + 1.79i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + (-0.363 - 0.264i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.650153512648029783500191470517, −9.203329565004427862446320380133, −8.072593331323981931576134602527, −7.20959258549407680931478236138, −6.50204432637920954565446184088, −5.83102155390273703359496365076, −4.95142789076056640088028564907, −3.74485248872826714763310412976, −3.02786332832364239982579300763, −1.78664605539351039720319431582, 0.66966583342303999508011152864, 2.14234174700817006234191096069, 3.01461485110001188030480755620, 4.29448312626659230039205124519, 5.04661338969856389937770548211, 5.66562311782593089322184638169, 6.91065057841776562156464409997, 7.40608109839807093652079845130, 8.787943786369489800801257112760, 9.372954497224734800451296500931

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