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Properties

Label 2-1380-115.68-c1-0-10
Degree $2$
Conductor $1380$
Sign $0.994 + 0.107i$
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 + (2.09 + 0.768i)5-s + (0.0117 + 0.0117i)7-s + 1.00i·9-s + 1.72i·11-s + (−2.37 − 2.37i)13-s + (−0.941 − 2.02i)15-s + (−1.66 − 1.66i)17-s + 6.73·19-s − 0.0165i·21-s + (4.79 − 0.0541i)23-s + (3.81 + 3.22i)25-s + (0.707 − 0.707i)27-s + 0.894i·29-s + 6.90·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.939 + 0.343i)5-s + (0.00443 + 0.00443i)7-s + 0.333i·9-s + 0.520i·11-s + (−0.657 − 0.657i)13-s + (−0.243 − 0.523i)15-s + (−0.404 − 0.404i)17-s + 1.54·19-s − 0.00361i·21-s + (0.999 − 0.0112i)23-s + (0.763 + 0.645i)25-s + (0.136 − 0.136i)27-s + 0.166i·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.994 + 0.107i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.753599248\)
\(L(\frac12)\) \(\approx\) \(1.753599248\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 - 0.768i)T \)
23 \( 1 + (-4.79 + 0.0541i)T \)
good7 \( 1 + (-0.0117 - 0.0117i)T + 7iT^{2} \)
11 \( 1 - 1.72iT - 11T^{2} \)
13 \( 1 + (2.37 + 2.37i)T + 13iT^{2} \)
17 \( 1 + (1.66 + 1.66i)T + 17iT^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
29 \( 1 - 0.894iT - 29T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 + (-0.0575 - 0.0575i)T + 37iT^{2} \)
41 \( 1 - 0.278T + 41T^{2} \)
43 \( 1 + (4.31 - 4.31i)T - 43iT^{2} \)
47 \( 1 + (7.61 - 7.61i)T - 47iT^{2} \)
53 \( 1 + (-9.64 + 9.64i)T - 53iT^{2} \)
59 \( 1 - 15.1iT - 59T^{2} \)
61 \( 1 + 5.91iT - 61T^{2} \)
67 \( 1 + (-1.99 - 1.99i)T + 67iT^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-7.55 - 7.55i)T + 73iT^{2} \)
79 \( 1 + 3.91T + 79T^{2} \)
83 \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 + (9.80 + 9.80i)T + 97iT^{2} \)
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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

−9.839487415448481555643011882927, −8.853503077405349369008616659929, −7.74662620816615953588345054262, −7.03867115220242534897532581420, −6.37705143909967770718672965717, −5.29482773005270944127581883099, −4.89061992073010942104904167026, −3.19658390264068415957685094500, −2.34312055574950667296944077390, −1.04830715562270762871612926880, 1.00672766938372658778438746687, 2.37180769626786954754545335271, 3.53253653068674641684607106560, 4.80606680192288946448211818705, 5.27166218704857624789685266070, 6.26770570558767776744425080476, 6.93345230070992315331378158549, 8.097992343801898837662233881728, 9.050477280398807382520215468945, 9.569040731055050843244449401051

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