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Properties

Label 2-1380-92.91-c1-0-13
Degree $2$
Conductor $1380$
Sign $0.777 - 0.628i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (0.686 − 1.23i)2-s − i·3-s + (−1.05 − 1.69i)4-s + i·5-s + (−1.23 − 0.686i)6-s − 4.16·7-s + (−2.82 + 0.138i)8-s − 9-s + (1.23 + 0.686i)10-s − 0.270·11-s + (−1.69 + 1.05i)12-s + 4.42·13-s + (−2.86 + 5.14i)14-s + 15-s + (−1.76 + 3.58i)16-s + 5.99i·17-s + ⋯
L(s)  = 1  + (0.485 − 0.874i)2-s − 0.577i·3-s + (−0.528 − 0.849i)4-s + 0.447i·5-s + (−0.504 − 0.280i)6-s − 1.57·7-s + (−0.998 + 0.0490i)8-s − 0.333·9-s + (0.390 + 0.217i)10-s − 0.0814·11-s + (−0.490 + 0.304i)12-s + 1.22·13-s + (−0.764 + 1.37i)14-s + 0.258·15-s + (−0.442 + 0.896i)16-s + 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6527543343\)
\(L(\frac12)\) \(\approx\) \(0.6527543343\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 + 1.23i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (0.591 + 4.75i)T \)
good7 \( 1 + 4.16T + 7T^{2} \)
11 \( 1 + 0.270T + 11T^{2} \)
13 \( 1 - 4.42T + 13T^{2} \)
17 \( 1 - 5.99iT - 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
29 \( 1 + 3.66T + 29T^{2} \)
31 \( 1 - 8.52iT - 31T^{2} \)
37 \( 1 - 6.85iT - 37T^{2} \)
41 \( 1 + 0.587T + 41T^{2} \)
43 \( 1 - 8.40T + 43T^{2} \)
47 \( 1 - 3.72iT - 47T^{2} \)
53 \( 1 - 1.59iT - 53T^{2} \)
59 \( 1 - 0.0658iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 2.90iT - 89T^{2} \)
97 \( 1 - 2.29iT - 97T^{2} \)
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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

−9.918019054802075142598028582793, −8.905676272663860227042180333067, −8.322895971162628784164833763633, −6.82677097652908341924912826172, −6.29551987615712380962516230404, −5.75522990056964932207162563187, −4.21115459109729001252810540091, −3.44566619680207906526157766976, −2.66013403673506655496052445237, −1.39550165743618932308841613395, 0.22922008770183549429851899954, 2.74520341533600649690172759933, 3.70110421128806702024581867622, 4.29035712916100079484329720435, 5.61240802201079084130901106905, 5.93992321123072266194801598276, 6.96161904928024191226025506746, 7.73686859481422131158567777837, 8.908085503695733710232308727447, 9.236558591810662973439703359406

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