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Properties

Label 2-126-63.58-c3-0-18
Degree $2$
Conductor $126$
Sign $0.926 + 0.375i$
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  + 2·2-s + (3.90 − 3.43i)3-s + 4·4-s + (5.69 + 9.87i)5-s + (7.80 − 6.86i)6-s + (8.79 − 16.2i)7-s + 8·8-s + (3.43 − 26.7i)9-s + (11.3 + 19.7i)10-s + (−22.6 + 39.2i)11-s + (15.6 − 13.7i)12-s + (1.14 − 1.98i)13-s + (17.5 − 32.5i)14-s + (56.1 + 18.9i)15-s + 16·16-s + (15.9 + 27.5i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.750 − 0.660i)3-s + 0.5·4-s + (0.509 + 0.883i)5-s + (0.530 − 0.467i)6-s + (0.474 − 0.880i)7-s + 0.353·8-s + (0.127 − 0.991i)9-s + (0.360 + 0.624i)10-s + (−0.621 + 1.07i)11-s + (0.375 − 0.330i)12-s + (0.0244 − 0.0424i)13-s + (0.335 − 0.622i)14-s + (0.966 + 0.326i)15-s + 0.250·16-s + (0.227 + 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.926 + 0.375i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.25025 - 0.632884i\)
\(L(\frac12)\) \(\approx\) \(3.25025 - 0.632884i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (-3.90 + 3.43i)T \)
7 \( 1 + (-8.79 + 16.2i)T \)
good5 \( 1 + (-5.69 - 9.87i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (22.6 - 39.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-1.14 + 1.98i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-15.9 - 27.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.6 + 18.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (83.2 + 144. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-88.1 - 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 136.T + 2.97e4T^{2} \)
37 \( 1 + (167. - 289. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (96.6 - 167. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-44.4 - 76.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 512.T + 1.03e5T^{2} \)
53 \( 1 + (-222. - 384. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 114.T + 2.05e5T^{2} \)
61 \( 1 + 335.T + 2.26e5T^{2} \)
67 \( 1 + 572.T + 3.00e5T^{2} \)
71 \( 1 - 925.T + 3.57e5T^{2} \)
73 \( 1 + (322. + 557. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 555.T + 4.93e5T^{2} \)
83 \( 1 + (-424. - 736. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-459. + 796. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (297. + 515. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.03064167442304977485986635622, −12.14145746413321096407695509332, −10.69287356630264622554710149297, −9.977097039142991478384309149219, −8.222260926939521810221377843373, −7.19762970917274332844628763276, −6.43542591031831163687978231839, −4.63975173002378242447026935939, −3.11210548402064483228353510396, −1.82609919001879392963870865500, 2.04295688819627340975331732305, 3.52018365840006542631010104582, 5.09613440648027782401460827883, 5.67532222143494803682386399709, 7.82069311837573689888654479131, 8.735660205023677675445260867685, 9.689452982821288754833727161193, 10.99070320692697673419118920260, 12.06416847415098912272155195837, 13.26867999560487663963546574239

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