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Properties

Label 2-126-7.5-c2-0-2
Degree $2$
Conductor $126$
Sign $0.827 + 0.561i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
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 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (5.74 + 3.31i)5-s + (6.24 + 3.16i)7-s − 2.82·8-s + (8.12 − 4.68i)10-s + (−2.37 − 4.11i)11-s − 15.2i·13-s + (8.29 − 5.40i)14-s + (−2.00 + 3.46i)16-s + (3.25 − 1.88i)17-s + (3.62 + 2.09i)19-s − 13.2i·20-s − 6.72·22-s + (−13.8 + 24.0i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.14 + 0.663i)5-s + (0.891 + 0.452i)7-s − 0.353·8-s + (0.812 − 0.468i)10-s + (−0.216 − 0.374i)11-s − 1.17i·13-s + (0.592 − 0.386i)14-s + (−0.125 + 0.216i)16-s + (0.191 − 0.110i)17-s + (0.190 + 0.110i)19-s − 0.663i·20-s − 0.305·22-s + (−0.602 + 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.827 + 0.561i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88357 - 0.578766i\)
\(L(\frac12)\) \(\approx\) \(1.88357 - 0.578766i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-6.24 - 3.16i)T \)
good5 \( 1 + (-5.74 - 3.31i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.37 + 4.11i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 15.2iT - 169T^{2} \)
17 \( 1 + (-3.25 + 1.88i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.62 - 2.09i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (13.8 - 24.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 3.51T + 841T^{2} \)
31 \( 1 + (42.3 - 24.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.47 + 2.54i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 10.4T + 1.84e3T^{2} \)
47 \( 1 + (45.6 + 26.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-27.9 - 48.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (33.5 - 19.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (78.3 + 45.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-17.3 - 29.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 36.4T + 5.04e3T^{2} \)
73 \( 1 + (-45.5 + 26.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-16.8 + 29.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 127. iT - 6.88e3T^{2} \)
89 \( 1 + (-43.5 - 25.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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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

−13.12935617564820894871756620222, −11.98255477027408258499250476786, −10.88576858425717312980084696071, −10.19589713003998975617176384971, −9.041664726676451963914494701554, −7.66064020376770409011639782117, −5.93964998481566040784960370571, −5.23334897462403193014412667634, −3.22097014769224266594579225246, −1.84725363546139836055929860135, 1.86744176268768117033494795648, 4.31229353954624045612844459254, 5.28656042935089232232611565009, 6.49905413854969624875278392331, 7.78246684329286606170295815231, 8.945219059212668146450027977131, 9.901632775596043769530923478297, 11.27056577937482110114771811132, 12.49062685657304081116197587253, 13.44467507460793287790836326759

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