Properties

Label 2-1380-345.344-c1-0-29
Degree $2$
Conductor $1380$
Sign $-0.545 + 0.837i$
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  + (−1.63 + 0.577i)3-s + (−2.15 − 0.603i)5-s + 1.63·7-s + (2.33 − 1.88i)9-s − 0.207·11-s − 1.70i·13-s + (3.86 − 0.256i)15-s + 5.57i·17-s − 2.31i·19-s + (−2.67 + 0.945i)21-s + (−2.87 + 3.83i)23-s + (4.27 + 2.60i)25-s + (−2.72 + 4.42i)27-s − 4.70i·29-s − 2.76·31-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)3-s + (−0.962 − 0.270i)5-s + 0.618·7-s + (0.777 − 0.628i)9-s − 0.0624·11-s − 0.472i·13-s + (0.997 − 0.0663i)15-s + 1.35i·17-s − 0.532i·19-s + (−0.583 + 0.206i)21-s + (−0.600 + 0.799i)23-s + (0.854 + 0.520i)25-s + (−0.523 + 0.851i)27-s − 0.873i·29-s − 0.496·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.545 + 0.837i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.545 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4101294135\)
\(L(\frac12)\) \(\approx\) \(0.4101294135\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.577i)T \)
5 \( 1 + (2.15 + 0.603i)T \)
23 \( 1 + (2.87 - 3.83i)T \)
good7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 0.207T + 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 - 5.57iT - 17T^{2} \)
19 \( 1 + 2.31iT - 19T^{2} \)
29 \( 1 + 4.70iT - 29T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + 1.83T + 37T^{2} \)
41 \( 1 + 7.81iT - 41T^{2} \)
43 \( 1 + 3.48T + 43T^{2} \)
47 \( 1 - 6.72T + 47T^{2} \)
53 \( 1 + 4.72iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 8.69iT - 61T^{2} \)
67 \( 1 + 9.87T + 67T^{2} \)
71 \( 1 - 0.717iT - 71T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 4.32iT - 79T^{2} \)
83 \( 1 - 1.66iT - 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 9.51T + 97T^{2} \)
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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

−9.358930786826274771032564303279, −8.364782481738718460917559748416, −7.75010004966906852932637140199, −6.85043500748844969737090996240, −5.83196657440269562685872642066, −5.09233851277530695464763522521, −4.21174742592398237544745191506, −3.51289983824839386515510360429, −1.67316045459781342416066656154, −0.20983374543532406540198213137, 1.28667892960691057580520255922, 2.72775396974527720772861862103, 4.12453565293299828646902286682, 4.76805501016077609312051221461, 5.68943177166239849108906897786, 6.75213253457854833471812347669, 7.32596550936358642082166142814, 8.042571006944895149724728292510, 8.951797440923239627765579178520, 10.09868052757726126844558258148

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