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Properties

Label 2-126-7.2-c3-0-7
Degree $2$
Conductor $126$
Sign $-0.384 + 0.923i$
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 + (1.72 − 2.98i)5-s + (15.3 − 10.2i)7-s − 7.99·8-s + (−3.44 − 5.96i)10-s + (−18.0 − 31.2i)11-s + 10.2·13-s + (−2.44 − 36.9i)14-s + (−8 + 13.8i)16-s + (−59.2 − 102. i)17-s + (19.3 − 33.4i)19-s − 13.7·20-s − 72.2·22-s + (18.1 − 31.3i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.154 − 0.266i)5-s + (0.831 − 0.556i)7-s − 0.353·8-s + (−0.108 − 0.188i)10-s + (−0.495 − 0.857i)11-s + 0.218·13-s + (−0.0466 − 0.705i)14-s + (−0.125 + 0.216i)16-s + (−0.845 − 1.46i)17-s + (0.233 − 0.404i)19-s − 0.154·20-s − 0.700·22-s + (0.164 − 0.284i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.384 + 0.923i$
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: \((2,\ 126,\ (\ :3/2),\ -0.384 + 0.923i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.04690 - 1.56924i\)
\(L(\frac12)\) \(\approx\) \(1.04690 - 1.56924i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.3 + 10.2i)T \)
good5 \( 1 + (-1.72 + 2.98i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (18.0 + 31.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 10.2T + 2.19e3T^{2} \)
17 \( 1 + (59.2 + 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.3 + 33.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-18.1 + 31.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 12.1T + 2.43e4T^{2} \)
31 \( 1 + (-72.7 - 126. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-0.685 + 1.18i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168T + 6.89e4T^{2} \)
43 \( 1 - 299.T + 7.95e4T^{2} \)
47 \( 1 + (251. - 435. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-312. - 541. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-21.1 - 36.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-219. + 380. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-381. - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + (289. + 501. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (471. - 816. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 474.T + 5.71e5T^{2} \)
89 \( 1 + (410. - 711. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.10e3T + 9.12e5T^{2} \)
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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

−12.64289128613264325240324632283, −11.27770412684632610315727751451, −10.93383935914031998932593173468, −9.495509220873643605279895203765, −8.450285990871188726468792313369, −7.08448905188938241796665453448, −5.44842013434965770817595533861, −4.45524821723370248395692596107, −2.79034193057261186820444320465, −0.932528310475167938418237031541, 2.19341497681031660637575517025, 4.15690619668915615769109253788, 5.38329541463074062746574899800, 6.53513879232286599688583388775, 7.83090187460548258942125748805, 8.693883250173028703186761468593, 10.10488741618460695254025233936, 11.24773371364827481972147018536, 12.39611812527774152499686941524, 13.24942747897968474193502113748

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