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Properties

Label 2-126-7.4-c5-0-10
Degree $2$
Conductor $126$
Sign $0.954 + 0.297i$
Analytic cond. $20.2083$
Root an. cond. $4.49537$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−7.99 + 13.8i)4-s + (−18.0 − 31.2i)5-s + (−124. + 36.3i)7-s − 63.9·8-s + (72.2 − 125. i)10-s + (77.7 − 134. i)11-s + 1.15e3·13-s + (−374. − 358. i)14-s + (−128 − 221. i)16-s + (619. − 1.07e3i)17-s + (140. + 242. i)19-s + 577.·20-s + 621.·22-s + (−1.74e3 − 3.01e3i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.322 − 0.559i)5-s + (−0.959 + 0.280i)7-s − 0.353·8-s + (0.228 − 0.395i)10-s + (0.193 − 0.335i)11-s + 1.90·13-s + (−0.511 − 0.488i)14-s + (−0.125 − 0.216i)16-s + (0.519 − 0.899i)17-s + (0.0890 + 0.154i)19-s + 0.322·20-s + 0.273·22-s + (−0.686 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(20.2083\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :5/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.789623028\)
\(L(\frac12)\) \(\approx\) \(1.789623028\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 \)
7 \( 1 + (124. - 36.3i)T \)
good5 \( 1 + (18.0 + 31.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-77.7 + 134. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 1.15e3T + 3.71e5T^{2} \)
17 \( 1 + (-619. + 1.07e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-140. - 242. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.74e3 + 3.01e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.65e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.15e3 + 2.00e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.16e3 - 2.02e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 3.81e3T + 1.15e8T^{2} \)
43 \( 1 - 3.92e3T + 1.47e8T^{2} \)
47 \( 1 + (5.55e3 + 9.62e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (5.59e3 - 9.68e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.00e3 - 5.20e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.41e3 - 1.28e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.14e4 + 3.72e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.99e4T + 1.80e9T^{2} \)
73 \( 1 + (2.27e4 - 3.94e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (5.44e4 + 9.43e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 5.58e4T + 3.93e9T^{2} \)
89 \( 1 + (4.77e4 + 8.27e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.50e4T + 8.58e9T^{2} \)
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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

−12.53502821219226223925923688596, −11.66355282081039878166700022589, −10.21583563731212785247776953432, −8.888144916159138326303288564028, −8.222277676114117299381946737550, −6.64455013320640174418906258726, −5.84798816889132912082864506519, −4.34237478867083419460853062848, −3.12464870506333951945060323090, −0.67091566798370549350792443427, 1.26911281930328008928473804443, 3.20184930981132514541879436194, 3.95338641969075467186022087405, 5.83595394863605623180042522513, 6.79928812124636240709315717683, 8.314658687741777074140426404412, 9.618733974723532627034598011547, 10.56913916154320163783088146012, 11.40274678766531393515656973449, 12.54153597704296942846033617761

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