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Properties

Label 2-126-63.11-c2-0-9
Degree $2$
Conductor $126$
Sign $0.781 + 0.624i$
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 + (2.81 − 1.03i)3-s − 2.00·4-s + (6.63 + 3.83i)5-s + (−1.45 − 3.98i)6-s + (2.93 + 6.35i)7-s + 2.82i·8-s + (6.87 − 5.80i)9-s + (5.42 − 9.38i)10-s + (−8.62 + 4.98i)11-s + (−5.63 + 2.06i)12-s + (−5.46 − 9.46i)13-s + (8.98 − 4.15i)14-s + (22.6 + 3.95i)15-s + 4.00·16-s + (−24.5 − 14.1i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.939 − 0.343i)3-s − 0.500·4-s + (1.32 + 0.766i)5-s + (−0.242 − 0.664i)6-s + (0.419 + 0.907i)7-s + 0.353i·8-s + (0.763 − 0.645i)9-s + (0.542 − 0.938i)10-s + (−0.784 + 0.452i)11-s + (−0.469 + 0.171i)12-s + (−0.420 − 0.727i)13-s + (0.641 − 0.296i)14-s + (1.51 + 0.263i)15-s + 0.250·16-s + (−1.44 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.781 + 0.624i$
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: \((2,\ 126,\ (\ :1),\ 0.781 + 0.624i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.93879 - 0.679545i\)
\(L(\frac12)\) \(\approx\) \(1.93879 - 0.679545i\)
\(L(2)\) 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 \( 1 + (-2.81 + 1.03i)T \)
7 \( 1 + (-2.93 - 6.35i)T \)
good5 \( 1 + (-6.63 - 3.83i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.62 - 4.98i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.46 + 9.46i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (24.5 + 14.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.29 + 9.17i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.643 + 0.371i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (39.7 + 22.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 47.6T + 961T^{2} \)
37 \( 1 + (-12.6 - 21.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (32.2 - 18.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.0 + 19.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 29.8iT - 2.20e3T^{2} \)
53 \( 1 + (-26.8 - 15.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 58.4iT - 3.48e3T^{2} \)
61 \( 1 + 25.6T + 3.72e3T^{2} \)
67 \( 1 - 1.60T + 4.48e3T^{2} \)
71 \( 1 + 98.6iT - 5.04e3T^{2} \)
73 \( 1 + (-43.7 + 75.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 107.T + 6.24e3T^{2} \)
83 \( 1 + (-135. - 78.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (54.2 - 31.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-30.3 + 52.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.35388293453364283990693162082, −12.13074864326174976073443733723, −10.81532809165672408551695125801, −9.798146891507645924946423884965, −9.078791108822563367862381499348, −7.80765018599989634087393503209, −6.38413973735342764101065567861, −4.91411536668824961054184249550, −2.68900844853857592976955820031, −2.22040270032356029088871793165, 1.94235162508570197513757166649, 4.16792243074472895315942618053, 5.23536243629375797551728859297, 6.71331804467454635548116065705, 8.070575689343287973429911334999, 8.902146892128510185179056118315, 9.849803637262751693225703339780, 10.77123643650284474407453393123, 12.90300973863794288023680246747, 13.47297957925351061315986627244

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