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Properties

Label 2-1380-115.68-c1-0-23
Degree $2$
Conductor $1380$
Sign $-0.857 - 0.515i$
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 + (0.811 − 2.08i)5-s + (−3.03 − 3.03i)7-s + 1.00i·9-s − 0.215i·11-s + (−0.104 − 0.104i)13-s + (−2.04 + 0.899i)15-s + (−0.838 − 0.838i)17-s − 3.62·19-s + 4.28i·21-s + (1.87 − 4.41i)23-s + (−3.68 − 3.38i)25-s + (0.707 − 0.707i)27-s + 5.72i·29-s − 0.698·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.362 − 0.931i)5-s + (−1.14 − 1.14i)7-s + 0.333i·9-s − 0.0648i·11-s + (−0.0288 − 0.0288i)13-s + (−0.528 + 0.232i)15-s + (−0.203 − 0.203i)17-s − 0.832·19-s + 0.935i·21-s + (0.390 − 0.920i)23-s + (−0.736 − 0.676i)25-s + (0.136 − 0.136i)27-s + 1.06i·29-s − 0.125·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.857 - 0.515i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.857 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4979102428\)
\(L(\frac12)\) \(\approx\) \(0.4979102428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.811 + 2.08i)T \)
23 \( 1 + (-1.87 + 4.41i)T \)
good7 \( 1 + (3.03 + 3.03i)T + 7iT^{2} \)
11 \( 1 + 0.215iT - 11T^{2} \)
13 \( 1 + (0.104 + 0.104i)T + 13iT^{2} \)
17 \( 1 + (0.838 + 0.838i)T + 17iT^{2} \)
19 \( 1 + 3.62T + 19T^{2} \)
29 \( 1 - 5.72iT - 29T^{2} \)
31 \( 1 + 0.698T + 31T^{2} \)
37 \( 1 + (-3.51 - 3.51i)T + 37iT^{2} \)
41 \( 1 + 1.54T + 41T^{2} \)
43 \( 1 + (-1.20 + 1.20i)T - 43iT^{2} \)
47 \( 1 + (3.22 - 3.22i)T - 47iT^{2} \)
53 \( 1 + (0.708 - 0.708i)T - 53iT^{2} \)
59 \( 1 - 2.79iT - 59T^{2} \)
61 \( 1 - 5.64iT - 61T^{2} \)
67 \( 1 + (5.33 + 5.33i)T + 67iT^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-9.19 - 9.19i)T + 73iT^{2} \)
79 \( 1 + 0.634T + 79T^{2} \)
83 \( 1 + (-3.22 + 3.22i)T - 83iT^{2} \)
89 \( 1 + 9.91T + 89T^{2} \)
97 \( 1 + (11.7 + 11.7i)T + 97iT^{2} \)
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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

−9.109741009970871722330352976899, −8.369924820555330899222609656809, −7.31168417043640360261448445248, −6.62560870159789056352412865945, −5.93468281406274132808179960580, −4.82838654750630991279166006264, −4.08456323203344640347440192445, −2.82476635455439244207645583306, −1.34773615405423987050530699223, −0.21500584115852156018560241988, 2.13646539445015886141351099297, 3.02881600101299947915920288013, 3.93893073875073277224390667960, 5.25651183506350029416688854190, 6.09879053609311128299035539561, 6.46003554853379909509367540969, 7.48390323205879758286189213606, 8.661278382793318805466914775257, 9.488956905112277414786187201673, 9.894723348599403549218252530328

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