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Properties

Label 2-1260-105.23-c1-0-13
Degree $2$
Conductor $1260$
Sign $-0.243 + 0.969i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (1.65 + 1.50i)5-s + (0.407 − 2.61i)7-s + (−3.45 + 1.99i)11-s + (−4.61 − 4.61i)13-s + (1.34 − 5.02i)17-s + (−4.70 − 2.71i)19-s + (−2.23 − 8.33i)23-s + (0.451 + 4.97i)25-s + 3.86·29-s + (−0.780 − 1.35i)31-s + (4.61 − 3.70i)35-s + (0.0737 + 0.275i)37-s + 3.13i·41-s + (2.78 + 2.78i)43-s + (0.765 − 0.204i)47-s + ⋯
L(s)  = 1  + (0.738 + 0.674i)5-s + (0.154 − 0.988i)7-s + (−1.04 + 0.601i)11-s + (−1.28 − 1.28i)13-s + (0.326 − 1.21i)17-s + (−1.07 − 0.622i)19-s + (−0.465 − 1.73i)23-s + (0.0902 + 0.995i)25-s + 0.716·29-s + (−0.140 − 0.242i)31-s + (0.780 − 0.625i)35-s + (0.0121 + 0.0452i)37-s + 0.490i·41-s + (0.423 + 0.423i)43-s + (0.111 − 0.0299i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.243 + 0.969i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.140445716\)
\(L(\frac12)\) \(\approx\) \(1.140445716\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.65 - 1.50i)T \)
7 \( 1 + (-0.407 + 2.61i)T \)
good11 \( 1 + (3.45 - 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.61 + 4.61i)T + 13iT^{2} \)
17 \( 1 + (-1.34 + 5.02i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.70 + 2.71i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 + 8.33i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3.86T + 29T^{2} \)
31 \( 1 + (0.780 + 1.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0737 - 0.275i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.13iT - 41T^{2} \)
43 \( 1 + (-2.78 - 2.78i)T + 43iT^{2} \)
47 \( 1 + (-0.765 + 0.204i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.99 - 1.60i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.19 - 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.70 + 1.52i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.668iT - 71T^{2} \)
73 \( 1 + (2.89 - 10.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (13.8 + 8.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.96 + 3.96i)T - 83iT^{2} \)
89 \( 1 + (-5.68 + 9.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.4 + 12.4i)T - 97iT^{2} \)
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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

−9.857159623727932589761710929135, −8.588411848014916857089841355077, −7.54584180408566522639250695134, −7.18460019356972422817001607722, −6.18819230137761901469076410010, −5.08789927099579343124910731275, −4.49373314012726786128800088655, −2.87532696353346382094425128985, −2.38451189515147677785544371281, −0.44044870749605563960501156091, 1.75734041973328855597840233946, 2.44387370772620386345042535656, 3.93488239422308145547173067582, 5.05751941075195426149959380526, 5.64693037430100762021383987580, 6.39250995541996798194984190825, 7.64063812741128767542842105704, 8.453977688762434634724647702554, 9.040463482286475083414965246810, 9.896474313448423513594537896965

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