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Properties

Label 2-126-63.23-c2-0-9
Degree $2$
Conductor $126$
Sign $0.902 + 0.429i$
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 + (1.98 + 2.24i)3-s − 2.00·4-s + (8.39 − 4.84i)5-s + (3.17 − 2.80i)6-s + (−3.70 + 5.93i)7-s + 2.82i·8-s + (−1.10 + 8.93i)9-s + (−6.85 − 11.8i)10-s + (−0.647 − 0.373i)11-s + (−3.97 − 4.49i)12-s + (8.35 − 14.4i)13-s + (8.39 + 5.24i)14-s + (27.5 + 9.25i)15-s + 4.00·16-s + (5.71 − 3.30i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.662 + 0.749i)3-s − 0.500·4-s + (1.67 − 0.969i)5-s + (0.529 − 0.468i)6-s + (−0.529 + 0.848i)7-s + 0.353i·8-s + (−0.123 + 0.992i)9-s + (−0.685 − 1.18i)10-s + (−0.0588 − 0.0339i)11-s + (−0.331 − 0.374i)12-s + (0.642 − 1.11i)13-s + (0.599 + 0.374i)14-s + (1.83 + 0.616i)15-s + 0.250·16-s + (0.336 − 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.902 + 0.429i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.902 + 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88690 - 0.426193i\)
\(L(\frac12)\) \(\approx\) \(1.88690 - 0.426193i\)
\(L(2)\) 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 \( 1 + (-1.98 - 2.24i)T \)
7 \( 1 + (3.70 - 5.93i)T \)
good5 \( 1 + (-8.39 + 4.84i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (0.647 + 0.373i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.35 + 14.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-5.71 + 3.30i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.429 - 0.744i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (16.5 - 9.55i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (19.2 - 11.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 39.3T + 961T^{2} \)
37 \( 1 + (34.2 - 59.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-27.4 - 15.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.3 + 52.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 1.19iT - 2.20e3T^{2} \)
53 \( 1 + (20.6 - 11.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 22.7iT - 3.48e3T^{2} \)
61 \( 1 + 8.34T + 3.72e3T^{2} \)
67 \( 1 + 16.1T + 4.48e3T^{2} \)
71 \( 1 + 36.6iT - 5.04e3T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 100.T + 6.24e3T^{2} \)
83 \( 1 + (-21.5 + 12.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-53.5 - 30.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-22.1 - 38.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.19623805334056854409376962597, −12.27605484579165833853511346900, −10.61578484990044220178740793360, −9.810559756364477545221805171560, −9.138226149508544950821277278832, −8.333944872201081140317971851387, −5.81882695353043874833980313751, −5.12835636302445973204359373132, −3.26868587445788478565358203080, −1.90182231787904765944716682602, 1.90684941401625106307590924035, 3.61839702999225409930274665357, 5.92200335834039016884443755942, 6.62890333715611425842573399099, 7.45992448292546473893464550012, 9.062663272122308881703725966378, 9.768730015772650557176046520225, 10.91201186675072983407404608243, 12.77066249363143557564908002420, 13.52890202404486568363264620061

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