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Properties

Label 2-126-63.13-c2-0-11
Degree $2$
Conductor $126$
Sign $0.988 - 0.152i$
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 + (1.54 − 2.57i)3-s + (−0.999 + 1.73i)4-s + (2.31 + 1.33i)5-s + (4.24 + 0.0754i)6-s + (6.01 − 3.58i)7-s − 2.82·8-s + (−4.22 − 7.94i)9-s + 3.78i·10-s + (4.28 + 7.42i)11-s + (2.90 + 5.24i)12-s + (4.93 + 2.85i)13-s + (8.63 + 4.83i)14-s + (7.02 − 3.88i)15-s + (−2.00 − 3.46i)16-s − 13.4i·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.515 − 0.856i)3-s + (−0.249 + 0.433i)4-s + (0.463 + 0.267i)5-s + (0.706 + 0.0125i)6-s + (0.859 − 0.511i)7-s − 0.353·8-s + (−0.468 − 0.883i)9-s + 0.378i·10-s + (0.389 + 0.675i)11-s + (0.242 + 0.437i)12-s + (0.379 + 0.219i)13-s + (0.617 + 0.345i)14-s + (0.468 − 0.259i)15-s + (−0.125 − 0.216i)16-s − 0.793i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.988 - 0.152i$
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: \((2,\ 126,\ (\ :1),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08436 + 0.159352i\)
\(L(\frac12)\) \(\approx\) \(2.08436 + 0.159352i\)
\(L(2)\) 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 \( 1 + (-1.54 + 2.57i)T \)
7 \( 1 + (-6.01 + 3.58i)T \)
good5 \( 1 + (-2.31 - 1.33i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-4.28 - 7.42i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.93 - 2.85i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 3.36iT - 361T^{2} \)
23 \( 1 + (19.7 - 34.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.4 - 38.8i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (45.9 + 26.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 43.8T + 1.36e3T^{2} \)
41 \( 1 + (24.3 + 14.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (5.62 + 9.73i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.7 - 14.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 20.0T + 2.80e3T^{2} \)
59 \( 1 + (-24.3 - 14.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-50.3 + 29.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5.55 - 9.61i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 56.3T + 5.04e3T^{2} \)
73 \( 1 - 101. iT - 5.32e3T^{2} \)
79 \( 1 + (-0.203 - 0.352i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-1.67 + 0.966i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 92.1iT - 7.92e3T^{2} \)
97 \( 1 + (-126. + 72.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.54198370552727144520359170335, −12.32894317101352243164142088659, −11.39794550456484844829785308666, −9.811434988606735943385664751047, −8.628463314391226679765228903770, −7.51663492930775692504051444739, −6.78894834201087322847156370952, −5.41398690303843292089979213307, −3.76150592922521567804025583583, −1.82840350527395716187848310015, 2.02064536954079683327626954164, 3.64216224229991314215047793021, 4.91433734782219694798583664553, 5.98543555193410989254806155135, 8.279771976848457608759124429045, 8.896747943201476565464296360402, 10.13660642293201501217372626946, 10.96670788455245078054903277624, 11.97689551896162769827197398661, 13.24657916373902058651817937353

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