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Properties

Label 2-126-63.25-c1-0-5
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 + (1.71 + 0.272i)3-s + 4-s + (1.59 − 2.75i)5-s + (−1.71 − 0.272i)6-s + (−2.56 + 0.658i)7-s − 8-s + (2.85 + 0.931i)9-s + (−1.59 + 2.75i)10-s + (−1.59 − 2.75i)11-s + (1.71 + 0.272i)12-s + (2.85 + 4.93i)13-s + (2.56 − 0.658i)14-s + (3.47 − 4.28i)15-s + 16-s + (−0.760 + 1.31i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.987 + 0.157i)3-s + 0.5·4-s + (0.711 − 1.23i)5-s + (−0.698 − 0.111i)6-s + (−0.968 + 0.249i)7-s − 0.353·8-s + (0.950 + 0.310i)9-s + (−0.503 + 0.871i)10-s + (−0.479 − 0.830i)11-s + (0.493 + 0.0785i)12-s + (0.790 + 1.36i)13-s + (0.684 − 0.176i)14-s + (0.896 − 1.10i)15-s + 0.250·16-s + (−0.184 + 0.319i)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

\(L(1)\) \(\approx\) \(1.06685 - 0.162213i\)
\(L(\frac12)\) \(\approx\) \(1.06685 - 0.162213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.272i)T \)
7 \( 1 + (2.56 - 0.658i)T \)
good5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.85 - 4.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.760 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.42T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.13T + 79T^{2} \)
83 \( 1 + (4.03 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.112 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.42 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.26923284259440935553396578644, −12.62201042515457502256080799295, −11.02414798299731951546670477714, −9.731785379738562141329801269055, −8.991248420725262629967408192999, −8.558379471628900280370049802263, −6.93677617308514214511416834780, −5.54941311319682606125326317940, −3.67301402463892483769100570211, −1.85617955315654329620368695849, 2.35854856132667627291828329605, 3.44125436550772866138974481360, 6.07947531546242995099900252362, 7.09042613394326803308586615805, 8.006248731667418759758946419130, 9.473391268741702771582022806324, 10.07768264473451158633234779635, 10.87706633430871348371638007932, 12.74398658146481157262066778976, 13.35903265525074886651046766527

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