Properties

Label 2-1380-5.4-c1-0-14
Degree $2$
Conductor $1380$
Sign $-0.241 + 0.970i$
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  i·3-s + (−0.539 + 2.17i)5-s i·7-s − 9-s − 0.539·11-s − 2.24i·13-s + (2.17 + 0.539i)15-s − 3.24i·17-s − 2.24·19-s − 21-s i·23-s + (−4.41 − 2.34i)25-s + i·27-s + 8.51·29-s − 6.70·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.241 + 0.970i)5-s − 0.377i·7-s − 0.333·9-s − 0.162·11-s − 0.623i·13-s + (0.560 + 0.139i)15-s − 0.787i·17-s − 0.515·19-s − 0.218·21-s − 0.208i·23-s + (−0.883 − 0.468i)25-s + 0.192i·27-s + 1.58·29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086815836\)
\(L(\frac12)\) \(\approx\) \(1.086815836\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (0.539 - 2.17i)T \)
23 \( 1 + iT \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 0.539T + 11T^{2} \)
13 \( 1 + 2.24iT - 13T^{2} \)
17 \( 1 + 3.24iT - 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
29 \( 1 - 8.51T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 + 7.04iT - 37T^{2} \)
41 \( 1 - 5.14T + 41T^{2} \)
43 \( 1 + 4.97iT - 43T^{2} \)
47 \( 1 + 8.87iT - 47T^{2} \)
53 \( 1 + 4.80iT - 53T^{2} \)
59 \( 1 - 0.170T + 59T^{2} \)
61 \( 1 + 2.67T + 61T^{2} \)
67 \( 1 + 8.31iT - 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 + 5.57T + 79T^{2} \)
83 \( 1 - 0.411iT - 83T^{2} \)
89 \( 1 - 6.68T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−9.324527269412785750462718811099, −8.388149110098643522922761839408, −7.53696894403896182796415224550, −7.04176226391883817887593943588, −6.21424706339756113769281520767, −5.27850682712706240461614187987, −4.05512352955260696291153529710, −3.06397088074925791257494830908, −2.16794299569408443005522095904, −0.45367968493702258984870384506, 1.40342519190880397215528845339, 2.77792864161789050613355794830, 4.06812720047121556237899066177, 4.60618732892329542147131398640, 5.58516488865258406035333239840, 6.34875831981042300438853456709, 7.56700315741502744625235129816, 8.450208899562433141567351153714, 8.930105506381315095964394945510, 9.704422680060286559147622711129

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