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Properties

Label 2-1380-115.22-c1-0-13
Degree $2$
Conductor $1380$
Sign $0.877 + 0.479i$
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  + (0.707 − 0.707i)3-s + (−2.23 + 0.146i)5-s + (2.93 − 2.93i)7-s − 1.00i·9-s + 6.31i·11-s + (0.106 − 0.106i)13-s + (−1.47 + 1.68i)15-s + (−2.74 + 2.74i)17-s + 7.24·19-s − 4.14i·21-s + (4.53 + 1.57i)23-s + (4.95 − 0.654i)25-s + (−0.707 − 0.707i)27-s − 9.94i·29-s + 6.76·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.997 + 0.0655i)5-s + (1.10 − 1.10i)7-s − 0.333i·9-s + 1.90i·11-s + (0.0295 − 0.0295i)13-s + (−0.380 + 0.434i)15-s + (−0.666 + 0.666i)17-s + 1.66·19-s − 0.905i·21-s + (0.944 + 0.327i)23-s + (0.991 − 0.130i)25-s + (−0.136 − 0.136i)27-s − 1.84i·29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892486229\)
\(L(\frac12)\) \(\approx\) \(1.892486229\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.23 - 0.146i)T \)
23 \( 1 + (-4.53 - 1.57i)T \)
good7 \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \)
11 \( 1 - 6.31iT - 11T^{2} \)
13 \( 1 + (-0.106 + 0.106i)T - 13iT^{2} \)
17 \( 1 + (2.74 - 2.74i)T - 17iT^{2} \)
19 \( 1 - 7.24T + 19T^{2} \)
29 \( 1 + 9.94iT - 29T^{2} \)
31 \( 1 - 6.76T + 31T^{2} \)
37 \( 1 + (1.65 - 1.65i)T - 37iT^{2} \)
41 \( 1 - 8.29T + 41T^{2} \)
43 \( 1 + (5.22 + 5.22i)T + 43iT^{2} \)
47 \( 1 + (3.00 + 3.00i)T + 47iT^{2} \)
53 \( 1 + (8.34 + 8.34i)T + 53iT^{2} \)
59 \( 1 - 1.05iT - 59T^{2} \)
61 \( 1 + 1.36iT - 61T^{2} \)
67 \( 1 + (4.30 - 4.30i)T - 67iT^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (5.69 - 5.69i)T - 73iT^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + (-4.73 - 4.73i)T + 83iT^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-3.23 + 3.23i)T - 97iT^{2} \)
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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

−9.593421940074517177053960893574, −8.424292617176447802876360150184, −7.71121041715577285276926503195, −7.36885206681959874378820614480, −6.61695184868743811207151897523, −4.94192732624595880485529706351, −4.43571226256652684793098689868, −3.55228923766432530612235131471, −2.15445213139062810575499014822, −1.00721913506444333475353437949, 1.08117091681771733483664105340, 2.86062007259225195784532814221, 3.32114114206508344282047599708, 4.76350464238126601501393548223, 5.16284936665971491078140157032, 6.29299181154327174767168459402, 7.54001566139146637558953948512, 8.138576162528427380059270121939, 8.908566497616147895339867624183, 9.168165137929518631438779524547

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