Properties

Label 2-126-63.25-c1-0-4
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: $\chi_{126} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863064 - 0.405157i\)
\(L(\frac12)\) \(\approx\) \(0.863064 - 0.405157i\)
\(L(\frac{3}{2})\) 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

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

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