Properties

Label 2-126-63.58-c1-0-0
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 + (0.933 + 1.45i)3-s + 4-s + (−0.296 − 0.514i)5-s + (−0.933 − 1.45i)6-s + (2.32 + 1.26i)7-s − 8-s + (−1.25 + 2.72i)9-s + (0.296 + 0.514i)10-s + (0.296 − 0.514i)11-s + (0.933 + 1.45i)12-s + (−1.25 + 2.17i)13-s + (−2.32 − 1.26i)14-s + (0.472 − 0.912i)15-s + 16-s + (1.46 + 2.52i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.538 + 0.842i)3-s + 0.5·4-s + (−0.132 − 0.229i)5-s + (−0.381 − 0.595i)6-s + (0.878 + 0.478i)7-s − 0.353·8-s + (−0.419 + 0.907i)9-s + (0.0938 + 0.162i)10-s + (0.0894 − 0.154i)11-s + (0.269 + 0.421i)12-s + (−0.348 + 0.603i)13-s + (−0.621 − 0.338i)14-s + (0.122 − 0.235i)15-s + 0.250·16-s + (0.354 + 0.613i)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: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863064 + 0.405157i\)
\(L(\frac12)\) \(\approx\) \(0.863064 + 0.405157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.933 - 1.45i)T \)
7 \( 1 + (-2.32 - 1.26i)T \)
good5 \( 1 + (0.296 + 0.514i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.296 + 0.514i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.46 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.86T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + (-4.02 - 6.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.64T + 59T^{2} \)
61 \( 1 + 6.64T + 61T^{2} \)
67 \( 1 + 1.91T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 - 6.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.24T + 79T^{2} \)
83 \( 1 + (-3.85 - 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.21 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.86 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.79669832597243754337526519283, −12.20719165931778426288293864829, −11.24776152228810882833312941281, −10.30791340036603909834077817022, −9.123207967030880575190222058389, −8.519243602052107907506029404002, −7.39327235666563156657912512035, −5.54977942797210003983703452427, −4.20953944430714044671990300953, −2.33583682642769888899118225151, 1.55177684972531063012113657708, 3.36667418652079540664379115619, 5.56369873423253165324445189779, 7.31175292918449682689485407400, 7.62045858941007342609328917881, 8.849693736931540421121569971305, 9.996672597497212720149546362732, 11.22750559284703742612475185952, 12.09272900853330935809398248859, 13.23214841466398813589838160687

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