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Properties

Label 2-126-63.5-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.739 - 0.672i$
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  + i·2-s + (−0.734 + 1.56i)3-s − 4-s + (0.483 + 0.837i)5-s + (−1.56 − 0.734i)6-s + (−2.16 + 1.52i)7-s − i·8-s + (−1.92 − 2.30i)9-s + (−0.837 + 0.483i)10-s + (4.82 + 2.78i)11-s + (0.734 − 1.56i)12-s + (−3.76 − 2.17i)13-s + (−1.52 − 2.16i)14-s + (−1.66 + 0.143i)15-s + 16-s + (1.97 + 3.41i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.424 + 0.905i)3-s − 0.5·4-s + (0.216 + 0.374i)5-s + (−0.640 − 0.299i)6-s + (−0.817 + 0.576i)7-s − 0.353i·8-s + (−0.640 − 0.768i)9-s + (−0.264 + 0.152i)10-s + (1.45 + 0.840i)11-s + (0.212 − 0.452i)12-s + (−1.04 − 0.603i)13-s + (−0.407 − 0.577i)14-s + (−0.431 + 0.0369i)15-s + 0.250·16-s + (0.478 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.739 - 0.672i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.739 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307576 + 0.795637i\)
\(L(\frac12)\) \(\approx\) \(0.307576 + 0.795637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.734 - 1.56i)T \)
7 \( 1 + (2.16 - 1.52i)T \)
good5 \( 1 + (-0.483 - 0.837i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.82 - 2.78i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.76 + 2.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.97 - 3.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.86 - 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 1.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.61 + 2.66i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (-0.243 + 0.421i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0818 + 0.141i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.35 + 7.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 + (1.74 - 1.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 + 5.17iT - 61T^{2} \)
67 \( 1 + 5.44T + 67T^{2} \)
71 \( 1 - 3.64iT - 71T^{2} \)
73 \( 1 + (2.15 - 1.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.60T + 79T^{2} \)
83 \( 1 + (4.20 + 7.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.05 - 3.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.2 + 5.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.28105280115845891351452331472, −12.53093511212808582499961568099, −11.94606865316832989959915827554, −10.19886994434164175131283764723, −9.748614245116468279832841445905, −8.642184013921248370950131425980, −6.93274013764479548307969008943, −6.05579374257893083108849066179, −4.82975891373926145996735156462, −3.34916023746122525681979123750, 1.07020872699685688886860239713, 3.12735342972144026663094198405, 4.93223164575374585997213808905, 6.40446363062036610696007359140, 7.40349651154147380643906204616, 9.023226972315540048908401047403, 9.792501654199271322495514112825, 11.34303738339882327738465729429, 11.85932891853816872888287900579, 12.93434603405141938096388799652

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