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Properties

Label 2-126-7.4-c1-0-1
Degree $2$
Conductor $126$
Sign $0.968 + 0.250i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + (2.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)10-s + (1.5 − 2.59i)11-s − 4·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)19-s − 3·20-s − 3·22-s + (−2 + 3.46i)25-s + (2 + 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (0.944 − 0.327i)7-s + 0.353·8-s + (0.474 − 0.821i)10-s + (0.452 − 0.783i)11-s − 1.10·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.458 + 0.794i)19-s − 0.670·20-s − 0.639·22-s + (−0.400 + 0.692i)25-s + (0.392 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01417 - 0.129240i\)
\(L(\frac12)\) \(\approx\) \(1.01417 - 0.129240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−13.45617691146784018129781377503, −12.01316835565085929352361181823, −11.14092985233623642253676869890, −10.35271728185846852446407742427, −9.399569344645120450055788494808, −7.994407119689783532594940018411, −6.93794384574737958812683295302, −5.41390844351482264400565507817, −3.60582205044088538160706793792, −2.04685327077511662941445831714, 1.75386172048355320554229063194, 4.73939802009150107229071478196, 5.34835803104314458794293990465, 6.97549366422668717138505761545, 8.151213123067926269352967750145, 9.194333467888403441282220988661, 9.849485421302246263818797824799, 11.45061696522902661556017684510, 12.48342779363780871045839587045, 13.48823057964127948742975721049

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