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Properties

Label 2-126-63.16-c1-0-4
Degree $2$
Conductor $126$
Sign $0.764 + 0.644i$
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 + (1.73 + 0.0789i)3-s + (−0.499 + 0.866i)4-s − 0.460·5-s + (−0.796 − 1.53i)6-s + (2.25 − 1.38i)7-s + 0.999·8-s + (2.98 + 0.273i)9-s + (0.230 + 0.398i)10-s − 3.64·11-s + (−0.933 + 1.45i)12-s + (0.730 + 1.26i)13-s + (−2.32 − 1.26i)14-s + (−0.796 − 0.0363i)15-s + (−0.5 − 0.866i)16-s + (−1.86 − 3.23i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.998 + 0.0455i)3-s + (−0.249 + 0.433i)4-s − 0.205·5-s + (−0.325 − 0.627i)6-s + (0.853 − 0.521i)7-s + 0.353·8-s + (0.995 + 0.0910i)9-s + (0.0728 + 0.126i)10-s − 1.09·11-s + (−0.269 + 0.421i)12-s + (0.202 + 0.350i)13-s + (−0.621 − 0.338i)14-s + (−0.205 − 0.00938i)15-s + (−0.125 − 0.216i)16-s + (−0.452 − 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10964 - 0.405525i\)
\(L(\frac12)\) \(\approx\) \(1.10964 - 0.405525i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0789i)T \)
7 \( 1 + (-2.25 + 1.38i)T \)
good5 \( 1 + 0.460T + 5T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
13 \( 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.02 - 3.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.13T + 23T^{2} \)
29 \( 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.257 + 0.445i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (6.62 + 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.32 + 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.59 - 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.38655916309380363200747855137, −12.24460914882675857656628456312, −10.99498002796629771214023868897, −10.21082096093572138453728571349, −8.985711004925294823556470709152, −8.060286130157985991655445873554, −7.23632067042690379541667706123, −4.88397518256203805059061476386, −3.58701670280910971974432199198, −1.97976735009963365560703596403, 2.28707844821845826024212305852, 4.28222483882954674924873337418, 5.72258534851055428004219357605, 7.37866297649901416844393703707, 8.164884061090475522182973982347, 8.915079072079775931538614957847, 10.17087262508644338245013821218, 11.26641607727892585447047360100, 12.84135577497650400002283873482, 13.57372873928981216857805813547

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