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Properties

Label 2-126-63.20-c1-0-7
Degree $2$
Conductor $126$
Sign $0.973 - 0.227i$
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 + (1.69 − 0.354i)3-s + (0.499 + 0.866i)4-s + (−0.895 − 1.55i)5-s + (1.64 + 0.541i)6-s + (−2.30 + 1.30i)7-s + 0.999i·8-s + (2.74 − 1.20i)9-s − 1.79i·10-s + (−2.07 − 1.20i)11-s + (1.15 + 1.29i)12-s + (−4.23 + 2.44i)13-s + (−2.64 − 0.0213i)14-s + (−2.06 − 2.31i)15-s + (−0.5 + 0.866i)16-s + 3.66·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.978 − 0.204i)3-s + (0.249 + 0.433i)4-s + (−0.400 − 0.693i)5-s + (0.671 + 0.220i)6-s + (−0.870 + 0.492i)7-s + 0.353i·8-s + (0.916 − 0.400i)9-s − 0.566i·10-s + (−0.627 − 0.362i)11-s + (0.333 + 0.372i)12-s + (−1.17 + 0.678i)13-s + (−0.707 − 0.00571i)14-s + (−0.533 − 0.596i)15-s + (−0.125 + 0.216i)16-s + 0.888·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.973 - 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64429 + 0.189241i\)
\(L(\frac12)\) \(\approx\) \(1.64429 + 0.189241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.354i)T \)
7 \( 1 + (2.30 - 1.30i)T \)
good5 \( 1 + (0.895 + 1.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.07 + 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.23 - 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 + (-3.26 + 1.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.68 + 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.02 - 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (4.04 + 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.48 - 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-7.29 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.81 - 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.285 + 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (1.51 - 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.00 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.74T + 89T^{2} \)
97 \( 1 + (-4.77 - 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.31134843902731998920980158617, −12.63761872732258452901319734511, −11.91160685226231937948591271754, −10.02929597041546836006660948411, −9.012759043153745240301433099487, −7.990865056316635269466516475209, −6.97976854512554160856058049292, −5.45759359419111951713949156966, −4.00557403492777075430877882652, −2.64462836175115182353216971708, 2.72762733790514778118175291117, 3.58700973574537418703642963439, 5.12123681218967616914593344974, 7.03094246054125165311454766198, 7.67249091853697528987286500293, 9.497407114699080719238687322785, 10.16222317985883652382836572157, 11.21037561514331702889042756707, 12.79763295853371021852264221670, 13.12630984219142605458705420489

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