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Properties

Label 2-126-63.58-c1-0-6
Degree $2$
Conductor $126$
Sign $0.381 + 0.924i$
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 + (−1.29 − 1.15i)3-s + 4-s + (−1.84 − 3.20i)5-s + (−1.29 − 1.15i)6-s + (2.64 − 0.0963i)7-s + 8-s + (0.349 + 2.97i)9-s + (−1.84 − 3.20i)10-s + (0.738 − 1.27i)11-s + (−1.29 − 1.15i)12-s + (−1.34 + 2.33i)13-s + (2.64 − 0.0963i)14-s + (−1.29 + 6.27i)15-s + 16-s + (3.28 + 5.69i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.747 − 0.664i)3-s + 0.5·4-s + (−0.827 − 1.43i)5-s + (−0.528 − 0.469i)6-s + (0.999 − 0.0364i)7-s + 0.353·8-s + (0.116 + 0.993i)9-s + (−0.584 − 1.01i)10-s + (0.222 − 0.385i)11-s + (−0.373 − 0.332i)12-s + (−0.374 + 0.648i)13-s + (0.706 − 0.0257i)14-s + (−0.334 + 1.62i)15-s + 0.250·16-s + (0.797 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01847 - 0.681620i\)
\(L(\frac12)\) \(\approx\) \(1.01847 - 0.681620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 1.15i)T \)
7 \( 1 + (-2.64 + 0.0963i)T \)
good5 \( 1 + (1.84 + 3.20i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.738 + 1.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.444 - 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.25 - 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.81T + 31T^{2} \)
37 \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.05 - 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 + (1.60 + 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.43 - 7.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.92137437493855910837355993710, −12.09097385397677620430892230161, −11.72185957136165199013414931367, −10.47594437391978726203764344284, −8.448713510524871189063991758767, −7.87737365368972064854697689364, −6.32830505874455246230444707695, −5.05737023458522022568593748150, −4.26703273620212417774205280082, −1.46141373801436768922324694872, 3.07300917069112844660040710051, 4.36834588794797656110852059494, 5.53604112642445751447978710296, 6.93747758323668509053848696241, 7.80386975282605281087470722572, 9.861889226592153505154053149997, 10.74665111443308044063767746057, 11.70800147508532354920174211554, 11.97436172904957858086888779989, 13.87465224765183144588727480505

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