Properties

Label 2-126-63.38-c1-0-2
Degree $2$
Conductor $126$
Sign $-0.135 - 0.990i$
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 + (1.52 + 0.816i)3-s − 4-s + (−1.82 + 3.15i)5-s + (−0.816 + 1.52i)6-s + (−1.58 − 2.12i)7-s i·8-s + (1.66 + 2.49i)9-s + (−3.15 − 1.82i)10-s + (4.38 − 2.53i)11-s + (−1.52 − 0.816i)12-s + (2.94 − 1.69i)13-s + (2.12 − 1.58i)14-s + (−5.35 + 3.33i)15-s + 16-s + (−0.774 + 1.34i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.881 + 0.471i)3-s − 0.5·4-s + (−0.814 + 1.41i)5-s + (−0.333 + 0.623i)6-s + (−0.598 − 0.801i)7-s − 0.353i·8-s + (0.555 + 0.831i)9-s + (−0.997 − 0.576i)10-s + (1.32 − 0.763i)11-s + (−0.440 − 0.235i)12-s + (0.816 − 0.471i)13-s + (0.566 − 0.422i)14-s + (−1.38 + 0.860i)15-s + 0.250·16-s + (−0.187 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.135 - 0.990i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778803 + 0.892190i\)
\(L(\frac12)\) \(\approx\) \(0.778803 + 0.892190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.816i)T \)
7 \( 1 + (1.58 + 2.12i)T \)
good5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.707 - 0.408i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.47 - 0.850i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.60 + 2.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.16iT - 31T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.01 - 1.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 + 5.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.74T + 47T^{2} \)
53 \( 1 + (11.4 + 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 2.45T + 67T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 - 2.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (-0.768 + 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.01 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 + 3.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.05111882161516943989519073300, −13.07613608866841005648949366450, −11.28025497954490714766993275954, −10.51586918344550917803218588527, −9.328468602792257801908490429056, −8.167162597304026923735905049187, −7.18980838850759167388703518729, −6.26640073765063446594360134079, −3.92526522846699080652196257011, −3.42678836646236090500192303548, 1.54214748193136969685118606353, 3.50400897618964010540468902494, 4.62843204363667675784611840737, 6.58696251405970989556172423234, 8.136291357364964223287514342215, 9.102375474969984466050912792543, 9.378667621038746717245474313105, 11.46137484846577398864684534776, 12.34797897122208150165855478618, 12.73777738853046009045712714294

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