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Properties

Label 2-126-21.17-c1-0-0
Degree $2$
Conductor $126$
Sign $0.821 - 0.569i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.358 + 0.621i)5-s + (2.62 + 0.358i)7-s + 0.999i·8-s + (−0.621 − 0.358i)10-s + (2.59 + 1.5i)11-s + 2.44i·13-s + (−2.44 + i)14-s + (−0.5 − 0.866i)16-s + (2.95 − 5.12i)17-s + (−5.12 + 2.95i)19-s + 0.717·20-s − 3·22-s + (−3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.160 + 0.277i)5-s + (0.990 + 0.135i)7-s + 0.353i·8-s + (−0.196 − 0.113i)10-s + (0.783 + 0.452i)11-s + 0.679i·13-s + (−0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (0.717 − 1.24i)17-s + (−1.17 + 0.678i)19-s + 0.160·20-s − 0.639·22-s + (−0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.821 - 0.569i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.821 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868743 + 0.271700i\)
\(L(\frac12)\) \(\approx\) \(0.868743 + 0.271700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 - 0.358i)T \)
good5 \( 1 + (-0.358 - 0.621i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 + 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 + (-2.95 - 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.03 - 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 + 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−13.86607376711173754908863808815, −12.13388182003623915313219001650, −11.42206667533199962951889883363, −10.20827555350745823817352543257, −9.242140803145329833134425611150, −8.116893152052058923129282897091, −7.07772438518680706746806924718, −5.84533253371431099379475615897, −4.33296126047713043764598076727, −1.96378941236481054100875427111, 1.60834260819349869734916631043, 3.70555517225141664449845033932, 5.33816727758436827108866961852, 6.87941292926266907766017284983, 8.286696477906000483278812320345, 8.842859915429263761776500206928, 10.36097866751794932791629908042, 11.00889591995896199800984117752, 12.17961734461077163123430508229, 13.04919587853276655503885343637

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