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Properties

Label 2-126-63.13-c2-0-3
Degree $2$
Conductor $126$
Sign $0.944 - 0.328i$
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  + (−0.707 − 1.22i)2-s + (−2.95 + 0.522i)3-s + (−0.999 + 1.73i)4-s + (−4.10 − 2.36i)5-s + (2.72 + 3.24i)6-s + (5.29 + 4.58i)7-s + 2.82·8-s + (8.45 − 3.08i)9-s + 6.70i·10-s + (3.85 + 6.67i)11-s + (2.04 − 5.63i)12-s + (13.7 + 7.91i)13-s + (1.86 − 9.72i)14-s + (13.3 + 4.85i)15-s + (−2.00 − 3.46i)16-s + 4.53i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.984 + 0.174i)3-s + (−0.249 + 0.433i)4-s + (−0.820 − 0.473i)5-s + (0.454 + 0.541i)6-s + (0.756 + 0.654i)7-s + 0.353·8-s + (0.939 − 0.343i)9-s + 0.670i·10-s + (0.350 + 0.607i)11-s + (0.170 − 0.469i)12-s + (1.05 + 0.608i)13-s + (0.133 − 0.694i)14-s + (0.890 + 0.323i)15-s + (−0.125 − 0.216i)16-s + 0.266i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.755950 + 0.127757i\)
\(L(\frac12)\) \(\approx\) \(0.755950 + 0.127757i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (2.95 - 0.522i)T \)
7 \( 1 + (-5.29 - 4.58i)T \)
good5 \( 1 + (4.10 + 2.36i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.85 - 6.67i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.7 - 7.91i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 4.53iT - 289T^{2} \)
19 \( 1 - 17.9iT - 361T^{2} \)
23 \( 1 + (-7.38 + 12.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.9 - 22.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-35.5 - 20.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 55.1T + 1.36e3T^{2} \)
41 \( 1 + (20.3 + 11.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.8 - 55.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.3 + 11.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 41.2T + 2.80e3T^{2} \)
59 \( 1 + (-69.7 - 40.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.59 - 2.65i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-55.9 + 96.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 90.1T + 5.04e3T^{2} \)
73 \( 1 + 108. iT - 5.32e3T^{2} \)
79 \( 1 + (-0.901 - 1.56i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (133. - 77.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 + (-132. + 76.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.50441280650677455028953223697, −12.10126306535329718265845593536, −11.26300242385690300542330847847, −10.34395875934864983712380319117, −8.972399135401778438319751741228, −8.073646923440611536618580725552, −6.54405923059410754578222012276, −4.97762274407181322299567372366, −3.96614298625602509043054871851, −1.41960959165505462833531384483, 0.803913425542783174097566294973, 3.95308806928518965308001803866, 5.32952071876223003832145177435, 6.61724374663410688861712750476, 7.51437155911144201673334460438, 8.507935751619033572823372828505, 10.19286460103755618324942035564, 11.17126980239361875527209545091, 11.59804388321331246031616026954, 13.23699450564209185322401014773

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