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Properties

Label 2-126-63.11-c2-0-10
Degree $2$
Conductor $126$
Sign $0.383 + 0.923i$
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  + 1.41i·2-s + (−2.92 + 0.673i)3-s − 2.00·4-s + (−1.51 − 0.875i)5-s + (−0.952 − 4.13i)6-s + (−1.24 − 6.88i)7-s − 2.82i·8-s + (8.09 − 3.93i)9-s + (1.23 − 2.14i)10-s + (3.90 − 2.25i)11-s + (5.84 − 1.34i)12-s + (−4.80 − 8.32i)13-s + (9.74 − 1.76i)14-s + (5.02 + 1.53i)15-s + 4.00·16-s + (0.491 + 0.283i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.974 + 0.224i)3-s − 0.500·4-s + (−0.303 − 0.175i)5-s + (−0.158 − 0.689i)6-s + (−0.178 − 0.984i)7-s − 0.353i·8-s + (0.899 − 0.437i)9-s + (0.123 − 0.214i)10-s + (0.355 − 0.205i)11-s + (0.487 − 0.112i)12-s + (−0.369 − 0.640i)13-s + (0.695 − 0.125i)14-s + (0.334 + 0.102i)15-s + 0.250·16-s + (0.0288 + 0.0166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.383 + 0.923i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.383 + 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.475672 - 0.317473i\)
\(L(\frac12)\) \(\approx\) \(0.475672 - 0.317473i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (2.92 - 0.673i)T \)
7 \( 1 + (1.24 + 6.88i)T \)
good5 \( 1 + (1.51 + 0.875i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.90 + 2.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.80 + 8.32i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-0.491 - 0.283i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.8 + 18.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (23.7 + 13.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-48.9 - 28.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 39.9T + 961T^{2} \)
37 \( 1 + (7.44 + 12.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (28.0 - 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (1.47 - 2.55i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 55.1iT - 2.20e3T^{2} \)
53 \( 1 + (-48.3 - 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 12.4iT - 3.48e3T^{2} \)
61 \( 1 - 79.5T + 3.72e3T^{2} \)
67 \( 1 - 0.284T + 4.48e3T^{2} \)
71 \( 1 + 8.92iT - 5.04e3T^{2} \)
73 \( 1 + (-4.02 + 6.97i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 97.6T + 6.24e3T^{2} \)
83 \( 1 + (19.5 + 11.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-47.1 + 27.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-38.0 + 65.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.91415800234825460340504990818, −12.01099218981049804388899374973, −10.72112461119753872179166371818, −9.990193304504993163505932495799, −8.534995277216347370854122517031, −7.22367897330509736641696374912, −6.37286760649039152616510997446, −4.99781138223083017493402228212, −3.94427652259014196984899455409, −0.44660572925221070778970465797, 1.95525746981746895125787397456, 3.98285813505381257888514629755, 5.40502837666509206504133642264, 6.54058180260035351171863790353, 8.005525533864441782043563186570, 9.426483818397213088583597784373, 10.34954122683250085391579897470, 11.65997095256047316827331095803, 11.96883904105510729419640042825, 12.91226580896050662015069146947

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