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Properties

Label 2-126-63.25-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.580 - 0.814i$
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 + 1.73i·3-s + 4-s + (−1.5 + 2.59i)5-s − 1.73i·6-s + (−2 − 1.73i)7-s − 8-s − 2.99·9-s + (1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + 1.73i·12-s + (0.5 + 0.866i)13-s + (2 + 1.73i)14-s + (−4.5 − 2.59i)15-s + 16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.999i·3-s + 0.5·4-s + (−0.670 + 1.16i)5-s − 0.707i·6-s + (−0.755 − 0.654i)7-s − 0.353·8-s − 0.999·9-s + (0.474 − 0.821i)10-s + (0.452 + 0.783i)11-s + 0.499i·12-s + (0.138 + 0.240i)13-s + (0.534 + 0.462i)14-s + (−1.16 − 0.670i)15-s + 0.250·16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.580 - 0.814i$
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: \((2,\ 126,\ (\ :1/2),\ -0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272364 + 0.528551i\)
\(L(\frac12)\) \(\approx\) \(0.272364 + 0.528551i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.15432927235022881694464068787, −12.40394875032453460772703553711, −11.28783340494978169916094368559, −10.41558855417257958818082274299, −9.871231791840324108513352673647, −8.563544964422620653554016335354, −7.24072744906191133014276375488, −6.31989551707620819368844786179, −4.19774518413936593886313699907, −3.08455353736824444039938934560, 0.813715947122070317671206201682, 3.02756476173093793568102837816, 5.31286683815536039308197177838, 6.61474407336360658553355825354, 7.77319976749745855321281860600, 8.807739756753312397387325094851, 9.364205306908958065473550705383, 11.44217736124206367817293983731, 11.78867811891303595147579852749, 12.97546810616720897735671154607

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