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Properties

Label 2-126-63.13-c2-0-15
Degree $2$
Conductor $126$
Sign $-0.681 + 0.731i$
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 + (−0.208 − 2.99i)3-s + (−0.999 + 1.73i)4-s + (−6.78 − 3.91i)5-s + (3.51 − 2.37i)6-s + (−6.99 − 0.117i)7-s − 2.82·8-s + (−8.91 + 1.25i)9-s − 11.0i·10-s + (−1.05 − 1.82i)11-s + (5.39 + 2.63i)12-s + (8.95 + 5.17i)13-s + (−4.80 − 8.65i)14-s + (−10.3 + 21.1i)15-s + (−2.00 − 3.46i)16-s − 9.44i·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0696 − 0.997i)3-s + (−0.249 + 0.433i)4-s + (−1.35 − 0.783i)5-s + (0.586 − 0.395i)6-s + (−0.999 − 0.0167i)7-s − 0.353·8-s + (−0.990 + 0.138i)9-s − 1.10i·10-s + (−0.0958 − 0.165i)11-s + (0.449 + 0.219i)12-s + (0.688 + 0.397i)13-s + (−0.343 − 0.618i)14-s + (−0.687 + 1.40i)15-s + (−0.125 − 0.216i)16-s − 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.681 + 0.731i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.681 + 0.731i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.235359 - 0.540992i\)
\(L(\frac12)\) \(\approx\) \(0.235359 - 0.540992i\)
\(L(2)\) 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 \( 1 + (0.208 + 2.99i)T \)
7 \( 1 + (6.99 + 0.117i)T \)
good5 \( 1 + (6.78 + 3.91i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (1.05 + 1.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.95 - 5.17i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 9.44iT - 289T^{2} \)
19 \( 1 + 7.01iT - 361T^{2} \)
23 \( 1 + (-22.7 + 39.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (9.93 + 17.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (5.21 + 3.01i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 40.4T + 1.36e3T^{2} \)
41 \( 1 + (52.9 + 30.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-34.3 - 59.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (59.5 - 34.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 64.4T + 2.80e3T^{2} \)
59 \( 1 + (-36.8 - 21.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-9.89 + 5.71i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-4.32 + 7.49i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 56.4T + 5.04e3T^{2} \)
73 \( 1 + 64.7iT - 5.32e3T^{2} \)
79 \( 1 + (57.0 + 98.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-86.5 + 49.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 90.9iT - 7.92e3T^{2} \)
97 \( 1 + (-39.5 + 22.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.80679759140593927987187940566, −12.09775079093962913258902270870, −11.11353854696511115998155051755, −9.051949641189077748926058554534, −8.282920164974280935219763094988, −7.19716489463174075057530445263, −6.30450561351871673224858798743, −4.74756056431581789369899475417, −3.26310675318811760872401286385, −0.36150125668843312492306624711, 3.35455187767922989153517544712, 3.67630489981536934047100281597, 5.38109768255845257653497927352, 6.86124509519731194118525457917, 8.379273333628136166906357818923, 9.640878768548014367259143262584, 10.63800952418814488283395270738, 11.29220038408751871864989556271, 12.27033117718911950236517285895, 13.44071015695536997354255765856

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