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Properties

Label 2-126-63.25-c1-0-6
Degree $2$
Conductor $126$
Sign $0.954 + 0.297i$
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 + (1.09 − 1.34i)3-s + 4-s + (−0.880 + 1.52i)5-s + (1.09 − 1.34i)6-s + (−0.710 + 2.54i)7-s + 8-s + (−0.619 − 2.93i)9-s + (−0.880 + 1.52i)10-s + (−3.06 − 5.30i)11-s + (1.09 − 1.34i)12-s + (−0.380 − 0.658i)13-s + (−0.710 + 2.54i)14-s + (1.09 + 2.84i)15-s + 16-s + (−3.42 + 5.92i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.629 − 0.776i)3-s + 0.5·4-s + (−0.393 + 0.681i)5-s + (0.445 − 0.549i)6-s + (−0.268 + 0.963i)7-s + 0.353·8-s + (−0.206 − 0.978i)9-s + (−0.278 + 0.482i)10-s + (−0.923 − 1.59i)11-s + (0.314 − 0.388i)12-s + (−0.105 − 0.182i)13-s + (−0.189 + 0.681i)14-s + (0.281 + 0.735i)15-s + 0.250·16-s + (−0.829 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.954 + 0.297i$
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: \((2,\ 126,\ (\ :1/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62862 - 0.247629i\)
\(L(\frac12)\) \(\approx\) \(1.62862 - 0.247629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 + 1.34i)T \)
7 \( 1 + (0.710 - 2.54i)T \)
good5 \( 1 + (0.880 - 1.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.06 + 5.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.380 + 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.42 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.210 + 0.364i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.66T + 47T^{2} \)
53 \( 1 + (0.112 - 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.98T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 6.78T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.30 - 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.81 - 3.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.35328518435051297578107125548, −12.51823101486302001210760394190, −11.52077453840115923139381566217, −10.49901597208054911886051985291, −8.719898772262750288637973234236, −7.964977524218746352693662501751, −6.56328901889907596491356894861, −5.71166129204748888227721673535, −3.52685007260981365350541557936, −2.54431991180380188749178154192, 2.72792928975173492036662640354, 4.48598043478844694603052868366, 4.78123518836355121981460972426, 7.00872472023068219161933604050, 7.937487437062083688340066777214, 9.418995285115585363223342006468, 10.24648976024143572916864508197, 11.41245684781757137238706951690, 12.67972576650323258844256291084, 13.46840473264451846111128174784

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