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Properties

Label 2-1260-105.2-c1-0-0
Degree $2$
Conductor $1260$
Sign $-0.985 + 0.167i$
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  + (−0.832 + 2.07i)5-s + (−0.0650 − 2.64i)7-s + (−1.79 + 1.03i)11-s + (−0.920 + 0.920i)13-s + (−1.55 − 0.416i)17-s + (1.78 + 1.03i)19-s + (−4.89 + 1.31i)23-s + (−3.61 − 3.45i)25-s + 1.58·29-s + (−4.97 − 8.61i)31-s + (5.54 + 2.06i)35-s + (−4.52 + 1.21i)37-s + 11.8i·41-s + (−6.62 + 6.62i)43-s + (−2.65 − 9.91i)47-s + ⋯
L(s)  = 1  + (−0.372 + 0.928i)5-s + (−0.0245 − 0.999i)7-s + (−0.540 + 0.312i)11-s + (−0.255 + 0.255i)13-s + (−0.376 − 0.100i)17-s + (0.409 + 0.236i)19-s + (−1.01 + 0.273i)23-s + (−0.722 − 0.691i)25-s + 0.294·29-s + (−0.893 − 1.54i)31-s + (0.936 + 0.349i)35-s + (−0.744 + 0.199i)37-s + 1.84i·41-s + (−1.00 + 1.00i)43-s + (−0.387 − 1.44i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.985 + 0.167i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.985 + 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04012389648\)
\(L(\frac12)\) \(\approx\) \(0.04012389648\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.832 - 2.07i)T \)
7 \( 1 + (0.0650 + 2.64i)T \)
good11 \( 1 + (1.79 - 1.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.920 - 0.920i)T - 13iT^{2} \)
17 \( 1 + (1.55 + 0.416i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.78 - 1.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.89 - 1.31i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 + (4.97 + 8.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.52 - 1.21i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (6.62 - 6.62i)T - 43iT^{2} \)
47 \( 1 + (2.65 + 9.91i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.55 - 5.81i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.46 + 7.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.41 + 7.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.99 + 7.45i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.25iT - 71T^{2} \)
73 \( 1 + (8.38 + 2.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.95 - 1.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.626 + 0.626i)T + 83iT^{2} \)
89 \( 1 + (2.79 - 4.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.71 + 8.71i)T + 97iT^{2} \)
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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

−9.951019873767519860329483271916, −9.720220985662221861644043508263, −8.166514065440098302616849192723, −7.68342212363707820104448807696, −6.89511052335391203930693732305, −6.17400527890805472642580517338, −4.90449123341207723342007150258, −3.99614813206728317234760967335, −3.14295455377904121981972232796, −1.90253811963768572363577463033, 0.01586109174509004900233697394, 1.74652883376586893679975580488, 2.95114055453460844221699216566, 4.08509165815120146634015379595, 5.23532112842559450581475353838, 5.56312262333854748808246902609, 6.82132587001391055059742956143, 7.79624231734098138059874878740, 8.635992236238821325218933559513, 8.968407948570021683528942103279

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