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Properties

Label 2-126-9.5-c2-0-2
Degree $2$
Conductor $126$
Sign $0.973 - 0.228i$
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  + (−1.22 + 0.707i)2-s + (2.87 − 0.856i)3-s + (0.999 − 1.73i)4-s + (2.30 + 1.32i)5-s + (−2.91 + 3.08i)6-s + (1.32 + 2.29i)7-s + 2.82i·8-s + (7.53 − 4.92i)9-s − 3.75·10-s + (−2.21 + 1.27i)11-s + (1.39 − 5.83i)12-s + (9.70 − 16.8i)13-s + (−3.24 − 1.87i)14-s + (7.75 + 1.84i)15-s + (−2.00 − 3.46i)16-s + 28.6i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.958 − 0.285i)3-s + (0.249 − 0.433i)4-s + (0.460 + 0.265i)5-s + (−0.485 + 0.513i)6-s + (0.188 + 0.327i)7-s + 0.353i·8-s + (0.836 − 0.547i)9-s − 0.375·10-s + (−0.200 + 0.116i)11-s + (0.115 − 0.486i)12-s + (0.746 − 1.29i)13-s + (−0.231 − 0.133i)14-s + (0.516 + 0.123i)15-s + (−0.125 − 0.216i)16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.973 - 0.228i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55216 + 0.179377i\)
\(L(\frac12)\) \(\approx\) \(1.55216 + 0.179377i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (-2.87 + 0.856i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good5 \( 1 + (-2.30 - 1.32i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.21 - 1.27i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.70 + 16.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 28.6iT - 289T^{2} \)
19 \( 1 - 10.6T + 361T^{2} \)
23 \( 1 + (-6.68 - 3.85i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (1.89 - 1.09i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (13.7 - 23.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 37.5T + 1.36e3T^{2} \)
41 \( 1 + (62.6 + 36.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (18.8 + 32.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (37.9 - 21.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 24.2iT - 2.80e3T^{2} \)
59 \( 1 + (69.5 + 40.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.15 - 10.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.0 - 64.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 98.3iT - 5.04e3T^{2} \)
73 \( 1 - 93.6T + 5.32e3T^{2} \)
79 \( 1 + (-47.0 - 81.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-35.7 + 20.6i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 75.2iT - 7.92e3T^{2} \)
97 \( 1 + (82.0 + 142. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.31641771161678244173683669711, −12.35268122790943916577122971744, −10.68750872511978381777309556973, −9.964499031159360250839093451700, −8.647971058301836894764123128960, −8.084003518033516545530656723603, −6.78121193263607762469865211094, −5.56284608123726320406529651701, −3.38313345182035940747577779378, −1.73330145354393589846069755357, 1.70940218848197798478506605751, 3.32010112285471887281677918219, 4.84087019396685149202482070385, 6.86001723003740154437405358314, 7.956792740295735838747569981556, 9.166595137025813065020053373099, 9.590424338350859863645851419991, 10.89392459419233190619235911813, 11.88817396667257449109963208404, 13.54418504803637613611403770463

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