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Properties

Label 2-126-63.20-c1-0-5
Degree $2$
Conductor $126$
Sign $0.955 + 0.296i$
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  + (0.866 + 0.5i)2-s + (−0.0967 − 1.72i)3-s + (0.499 + 0.866i)4-s + (−0.183 − 0.317i)5-s + (0.780 − 1.54i)6-s + (2.53 − 0.744i)7-s + 0.999i·8-s + (−2.98 + 0.334i)9-s − 0.366i·10-s + (0.579 + 0.334i)11-s + (1.44 − 0.948i)12-s + (−0.867 + 0.500i)13-s + (2.57 + 0.624i)14-s + (−0.531 + 0.347i)15-s + (−0.5 + 0.866i)16-s − 4.98·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.0558 − 0.998i)3-s + (0.249 + 0.433i)4-s + (−0.0819 − 0.141i)5-s + (0.318 − 0.631i)6-s + (0.959 − 0.281i)7-s + 0.353i·8-s + (−0.993 + 0.111i)9-s − 0.115i·10-s + (0.174 + 0.100i)11-s + (0.418 − 0.273i)12-s + (−0.240 + 0.138i)13-s + (0.687 + 0.166i)14-s + (−0.137 + 0.0897i)15-s + (−0.125 + 0.216i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.955 + 0.296i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.955 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42732 - 0.216380i\)
\(L(\frac12)\) \(\approx\) \(1.42732 - 0.216380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.0967 + 1.72i)T \)
7 \( 1 + (-2.53 + 0.744i)T \)
good5 \( 1 + (0.183 + 0.317i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.579 - 0.334i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.867 - 0.500i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.98T + 17T^{2} \)
19 \( 1 - 6.35iT - 19T^{2} \)
23 \( 1 + (6.66 - 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.47 + 3.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + (2.15 + 3.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.16 + 7.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-4.36 - 7.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.29 + 2.47i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.44 - 9.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 + 4.07iT - 73T^{2} \)
79 \( 1 + (4.17 - 7.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.50 + 14.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (14.9 + 8.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.57288316139292417383944280641, −12.21171192149645490747745527602, −11.79281569709233964314183099210, −10.47976193392051250222653945074, −8.599984381479091275128703243443, −7.81127832244260575884512525793, −6.73316871150411389789454866224, −5.55860699066641321415793748218, −4.13966310637201910222760772019, −2.02400800871502868494148014425, 2.63237852966483782168630518529, 4.30730644030077758260086867199, 5.09896271639761607796918136948, 6.54276542348625188812231821558, 8.313696622102372043177262104453, 9.356704040157773154253201785871, 10.66240434570102001974334708735, 11.26070745610678108011031879522, 12.19369074462488241401915330835, 13.62509874943139790539041540332

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