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Properties

Label 2-126-7.2-c3-0-6
Degree $2$
Conductor $126$
Sign $-0.145 + 0.989i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (5.22 − 9.04i)5-s + (−12.3 + 13.7i)7-s + 7.99·8-s + (10.4 + 18.0i)10-s + (−30.5 − 52.9i)11-s − 59.2·13-s + (−11.4 − 35.2i)14-s + (−8 + 13.8i)16-s + (−10.2 − 17.7i)17-s + (40.1 − 69.5i)19-s − 41.7·20-s + 122.·22-s + (79.1 − 137. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.467 − 0.809i)5-s + (−0.669 + 0.743i)7-s + 0.353·8-s + (0.330 + 0.572i)10-s + (−0.837 − 1.45i)11-s − 1.26·13-s + (−0.218 − 0.672i)14-s + (−0.125 + 0.216i)16-s + (−0.145 − 0.252i)17-s + (0.485 − 0.840i)19-s − 0.467·20-s + 1.18·22-s + (0.717 − 1.24i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.145 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.436613 - 0.505542i\)
\(L(\frac12)\) \(\approx\) \(0.436613 - 0.505542i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 + (12.3 - 13.7i)T \)
good5 \( 1 + (-5.22 + 9.04i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.5 + 52.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 59.2T + 2.19e3T^{2} \)
17 \( 1 + (10.2 + 17.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.1 + 69.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-79.1 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 85.1T + 2.43e4T^{2} \)
31 \( 1 + (121. + 210. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (145. - 251. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 168T + 6.89e4T^{2} \)
43 \( 1 - 7.62T + 7.95e4T^{2} \)
47 \( 1 + (-84.6 + 146. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-125. - 216. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-402. - 697. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (16.5 - 28.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-138. - 240. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 631.T + 3.57e5T^{2} \)
73 \( 1 + (-384. - 665. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-209. + 362. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 761.T + 5.71e5T^{2} \)
89 \( 1 + (-786. + 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−12.92561439003126717312235273829, −11.61801107163965260911149908763, −10.24715213598626805680977377477, −9.179868164251782194389409565392, −8.561596535214405508747673469796, −7.14514331579886921539529247145, −5.77833682129597488886901496070, −5.01973198956340014473638086608, −2.72022205158737906360544173768, −0.36641668518863912306773905752, 2.07297364917298969944364265602, 3.45178931008482918989633262652, 5.11852399797098339356760306197, 6.96796977731882550833358806452, 7.56892226174014047775280635041, 9.472243527677998301842396325131, 10.08554675282715390342660813823, 10.78863651547897132630071226956, 12.26006799756190123874461406409, 12.95937448423676737248158434171

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