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Properties

Label 2-126-63.4-c3-0-19
Degree $2$
Conductor $126$
Sign $-0.339 + 0.940i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (0.828 − 5.12i)3-s + (−1.99 − 3.46i)4-s + 15.6·5-s + (−8.05 − 6.56i)6-s + (17.1 − 6.97i)7-s − 7.99·8-s + (−25.6 − 8.50i)9-s + (15.6 − 27.0i)10-s + 36.9·11-s + (−19.4 + 7.38i)12-s + (−17.5 + 30.3i)13-s + (5.08 − 36.6i)14-s + (12.9 − 80.2i)15-s + (−8 + 13.8i)16-s + (−41.2 + 71.4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.159 − 0.987i)3-s + (−0.249 − 0.433i)4-s + 1.39·5-s + (−0.548 − 0.446i)6-s + (0.926 − 0.376i)7-s − 0.353·8-s + (−0.949 − 0.314i)9-s + (0.494 − 0.856i)10-s + 1.01·11-s + (−0.467 + 0.177i)12-s + (−0.373 + 0.647i)13-s + (0.0970 − 0.700i)14-s + (0.223 − 1.38i)15-s + (−0.125 + 0.216i)16-s + (−0.588 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.339 + 0.940i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.339 + 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.45985 - 2.07985i\)
\(L(\frac12)\) \(\approx\) \(1.45985 - 2.07985i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (-0.828 + 5.12i)T \)
7 \( 1 + (-17.1 + 6.97i)T \)
good5 \( 1 - 15.6T + 125T^{2} \)
11 \( 1 - 36.9T + 1.33e3T^{2} \)
13 \( 1 + (17.5 - 30.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (41.2 - 71.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (51.0 + 88.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 106.T + 1.21e4T^{2} \)
29 \( 1 + (53.9 + 93.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-133. - 231. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-78.5 - 136. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-113. + 196. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-186. - 323. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-112. + 194. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (105. - 182. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (83.0 + 143. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (258. - 447. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-174. - 301. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 121.T + 3.57e5T^{2} \)
73 \( 1 + (-592. + 1.02e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (528. - 915. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-199. - 346. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (608. + 1.05e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (456. + 790. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.67599699290543166539836199440, −11.68361721203209235192531477354, −10.71186933771617199778731526709, −9.428522023335555431021749311106, −8.472101678304036085904259846735, −6.81577134729019947779401281119, −5.94570348427753064033486003769, −4.40374754943973899170298283619, −2.28752613253548916205560444141, −1.41781770073755426499050485516, 2.31594554433789812309554273962, 4.20818631812429034391525094366, 5.38102546846366558890025690766, 6.17815817446491313236826463093, 7.963387414897302617715895138112, 9.113682742550158002890637818173, 9.828026198054644488425946191021, 11.06648262184671870454038666364, 12.25125628788375682786937149885, 13.67980274498771167339148263011

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