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Properties

Label 2-126-63.32-c2-0-7
Degree $2$
Conductor $126$
Sign $0.161 - 0.986i$
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  + (1.22 + 0.707i)2-s + (0.878 + 2.86i)3-s + (0.999 + 1.73i)4-s − 1.75i·5-s + (−0.952 + 4.13i)6-s + (6.58 + 2.36i)7-s + 2.82i·8-s + (−7.45 + 5.04i)9-s + (1.23 − 2.14i)10-s − 4.51i·11-s + (−4.08 + 4.39i)12-s + (−4.80 + 8.32i)13-s + (6.39 + 7.55i)14-s + (5.02 − 1.53i)15-s + (−2.00 + 3.46i)16-s + (−0.491 − 0.283i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.292 + 0.956i)3-s + (0.249 + 0.433i)4-s − 0.350i·5-s + (−0.158 + 0.689i)6-s + (0.941 + 0.337i)7-s + 0.353i·8-s + (−0.828 + 0.560i)9-s + (0.123 − 0.214i)10-s − 0.410i·11-s + (−0.340 + 0.365i)12-s + (−0.369 + 0.640i)13-s + (0.456 + 0.539i)14-s + (0.334 − 0.102i)15-s + (−0.125 + 0.216i)16-s + (−0.0288 − 0.0166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.161 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64243 + 1.39500i\)
\(L(\frac12)\) \(\approx\) \(1.64243 + 1.39500i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-0.878 - 2.86i)T \)
7 \( 1 + (-6.58 - 2.36i)T \)
good5 \( 1 + 1.75iT - 25T^{2} \)
11 \( 1 + 4.51iT - 121T^{2} \)
13 \( 1 + (4.80 - 8.32i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (0.491 + 0.283i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.8 + 18.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 27.4iT - 529T^{2} \)
29 \( 1 + (-48.9 + 28.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.9 - 34.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (7.44 + 12.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (28.0 + 16.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (1.47 + 2.55i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (47.7 + 27.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (48.3 + 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (10.7 - 6.21i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.7 - 68.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (0.142 + 0.246i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 8.92iT - 5.04e3T^{2} \)
73 \( 1 + (-4.02 + 6.97i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (48.8 - 84.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (19.5 - 11.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (47.1 - 27.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-38.0 - 65.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.71766647112935537416749481074, −12.31349876735091099649006464955, −11.37147458020558733686430132417, −10.36764958737485209454740689596, −8.833843008022437008786334158169, −8.308530638350030540128302127981, −6.59433525584720389876712506289, −5.03549498822202310509064852832, −4.45023081359159921916959009281, −2.65944207844254330548409359149, 1.55274232557427973084301864643, 3.09328138814972004779601623895, 4.81833366274137620433455884526, 6.25045411266239198384260547757, 7.42854484187693403683677807063, 8.351724117082971192428048564890, 10.01047719595848467555359116164, 11.10249970251232111796900910921, 12.06362893952364987892812287767, 12.87389994464302541142800468259

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