Properties

Label 2-126-63.25-c1-0-7
Degree $2$
Conductor $126$
Sign $0.638 + 0.769i$
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  + 2-s + (−0.796 − 1.53i)3-s + 4-s + (0.230 − 0.398i)5-s + (−0.796 − 1.53i)6-s + (0.0665 − 2.64i)7-s + 8-s + (−1.73 + 2.45i)9-s + (0.230 − 0.398i)10-s + (1.82 + 3.15i)11-s + (−0.796 − 1.53i)12-s + (0.730 + 1.26i)13-s + (0.0665 − 2.64i)14-s + (−0.796 − 0.0363i)15-s + 16-s + (−1.86 + 3.23i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.460 − 0.887i)3-s + 0.5·4-s + (0.102 − 0.178i)5-s + (−0.325 − 0.627i)6-s + (0.0251 − 0.999i)7-s + 0.353·8-s + (−0.576 + 0.816i)9-s + (0.0728 − 0.126i)10-s + (0.549 + 0.952i)11-s + (−0.230 − 0.443i)12-s + (0.202 + 0.350i)13-s + (0.0177 − 0.706i)14-s + (−0.205 − 0.00938i)15-s + 0.250·16-s + (−0.452 + 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25289 - 0.588157i\)
\(L(\frac12)\) \(\approx\) \(1.25289 - 0.588157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.796 + 1.53i)T \)
7 \( 1 + (-0.0665 + 2.64i)T \)
good5 \( 1 + (-0.230 + 0.398i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.566 - 0.981i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.514T + 31T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 2.32T + 67T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.00T + 79T^{2} \)
83 \( 1 + (-3.32 + 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.36 + 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 - 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.07645101205628820733490364478, −12.52999222786622913520448501869, −11.30924353949242507582013765550, −10.55906878523478005745832733877, −8.891384912201471078918654645668, −7.27126321677831000274552548427, −6.78591588393007215246654466618, −5.31840217814920452606396244660, −3.99408284332197900928906549926, −1.77243743364827373841162369696, 2.92163865274782315884676932020, 4.33425611917981257235581876737, 5.68096524436360357168280617324, 6.35402688273999265460144535290, 8.339737646739737162255251476344, 9.435233616337103684132713714855, 10.65748041841555795282536783202, 11.56355889596084302289841119226, 12.28081182802122598219148596302, 13.64597801010632945500376093392

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