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Properties

Label 2-126-63.23-c2-0-7
Degree $2$
Conductor $126$
Sign $0.570 + 0.821i$
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 + (−2.24 + 1.98i)3-s − 2.00·4-s + (1.84 − 1.06i)5-s + (2.80 + 3.18i)6-s + (6.16 − 3.31i)7-s + 2.82i·8-s + (1.11 − 8.93i)9-s + (−1.50 − 2.60i)10-s + (7.75 + 4.47i)11-s + (4.49 − 3.97i)12-s + (10.4 − 18.0i)13-s + (−4.68 − 8.72i)14-s + (−2.03 + 6.05i)15-s + 4.00·16-s + (9.01 − 5.20i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.749 + 0.661i)3-s − 0.500·4-s + (0.368 − 0.212i)5-s + (0.468 + 0.530i)6-s + (0.881 − 0.473i)7-s + 0.353i·8-s + (0.123 − 0.992i)9-s + (−0.150 − 0.260i)10-s + (0.705 + 0.407i)11-s + (0.374 − 0.330i)12-s + (0.801 − 1.38i)13-s + (−0.334 − 0.622i)14-s + (−0.135 + 0.403i)15-s + 0.250·16-s + (0.530 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.570 + 0.821i$
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: \((2,\ 126,\ (\ :1),\ 0.570 + 0.821i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10126 - 0.576320i\)
\(L(\frac12)\) \(\approx\) \(1.10126 - 0.576320i\)
\(L(2)\) 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 \( 1 + (2.24 - 1.98i)T \)
7 \( 1 + (-6.16 + 3.31i)T \)
good5 \( 1 + (-1.84 + 1.06i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.75 - 4.47i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.4 + 18.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-9.01 + 5.20i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.15 + 10.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (16.5 - 9.55i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (28.3 - 16.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 37.8T + 961T^{2} \)
37 \( 1 + (-16.9 + 29.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (28.7 + 16.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-10.3 - 17.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 66.2iT - 2.20e3T^{2} \)
53 \( 1 + (-2.72 + 1.57i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 84.2iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + 66.7T + 4.48e3T^{2} \)
71 \( 1 - 115. iT - 5.04e3T^{2} \)
73 \( 1 + (19.6 + 33.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 52.4T + 6.24e3T^{2} \)
83 \( 1 + (-9.25 + 5.34i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (61.9 + 35.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-64.5 - 111. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.78469806046971236271341710586, −11.70157484311221586120910012197, −10.97079006492282661026864082030, −10.05744268565889010959241368527, −9.126556810022198323987754995280, −7.68360487261361136930867455065, −5.88374241666894215270412943534, −4.86062547685707077401047174702, −3.59155423673850803030814400671, −1.16791691819568940376444614424, 1.64969643893435849177214662094, 4.35691355855334114257210523855, 5.84278195595184119181079856445, 6.43193041833708757201129052969, 7.80675949797654481774675366991, 8.762429394620242112453965355732, 10.20143871050247893113495161513, 11.56442001031287976127342460966, 12.05276695096364287145625634688, 13.66998266780206505518524749961

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