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Properties

Label 2-126-7.2-c3-0-5
Degree $2$
Conductor $126$
Sign $0.701 + 0.712i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3 + 5.19i)5-s + (−3.5 − 18.1i)7-s + 7.99·8-s + (−6 − 10.3i)10-s + (−15 − 25.9i)11-s + 53·13-s + (35 + 12.1i)14-s + (−8 + 13.8i)16-s + (−42 − 72.7i)17-s + (48.5 − 84.0i)19-s + 24·20-s + 60·22-s + (42 − 72.7i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.268 + 0.464i)5-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (−0.189 − 0.328i)10-s + (−0.411 − 0.712i)11-s + 1.13·13-s + (0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.599 − 1.03i)17-s + (0.585 − 1.01i)19-s + 0.268·20-s + 0.581·22-s + (0.380 − 0.659i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.936180 - 0.392337i\)
\(L(\frac12)\) \(\approx\) \(0.936180 - 0.392337i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 + (3.5 + 18.1i)T \)
good5 \( 1 + (3 - 5.19i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15 + 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 53T + 2.19e3T^{2} \)
17 \( 1 + (42 + 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.5 + 84.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 180T + 2.43e4T^{2} \)
31 \( 1 + (89.5 + 155. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-72.5 + 125. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 126T + 6.89e4T^{2} \)
43 \( 1 + 325T + 7.95e4T^{2} \)
47 \( 1 + (183 - 316. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (384 + 665. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (132 + 228. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (409 - 708. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-261.5 - 452. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (-21.5 - 37.2i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-585.5 + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 810T + 5.71e5T^{2} \)
89 \( 1 + (300 - 519. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 386T + 9.12e5T^{2} \)
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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

−13.23631760972399076113305208127, −11.31224776686604811967310098881, −10.80006921978571499697118325398, −9.508046886470679068998888782265, −8.375620117802380598793634997412, −7.23929237673795837249758600062, −6.40915339964381588625444221209, −4.82475527453791270161547633151, −3.24119998891461775981616232013, −0.62679119020312122629829294592, 1.64331533452878247063824713395, 3.36612555773542702531939328324, 4.90463477642954608666329915241, 6.35924533062181775072023553217, 8.083101007813237997371852869290, 8.764141733591846507659728781155, 9.912975577999338355814206092532, 10.97029375619024527678803487266, 12.14110038804396856393985258513, 12.66417862033728567336785928234

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