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Properties

Label 2-126-63.23-c2-0-12
Degree $2$
Conductor $126$
Sign $0.489 + 0.871i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (0.146 − 2.99i)3-s − 2.00·4-s + (2.36 − 1.36i)5-s + (4.23 + 0.207i)6-s + (−2.54 − 6.52i)7-s − 2.82i·8-s + (−8.95 − 0.879i)9-s + (1.92 + 3.34i)10-s + (−2.52 − 1.45i)11-s + (−0.293 + 5.99i)12-s + (10.4 − 18.1i)13-s + (9.22 − 3.59i)14-s + (−3.73 − 7.27i)15-s + 4.00·16-s + (24.7 − 14.2i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0489 − 0.998i)3-s − 0.500·4-s + (0.472 − 0.272i)5-s + (0.706 + 0.0346i)6-s + (−0.363 − 0.931i)7-s − 0.353i·8-s + (−0.995 − 0.0977i)9-s + (0.192 + 0.334i)10-s + (−0.229 − 0.132i)11-s + (−0.0244 + 0.499i)12-s + (0.805 − 1.39i)13-s + (0.658 − 0.256i)14-s + (−0.249 − 0.485i)15-s + 0.250·16-s + (1.45 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.489 + 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.11839 - 0.654540i\)
\(L(\frac12)\) \(\approx\) \(1.11839 - 0.654540i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-0.146 + 2.99i)T \)
7 \( 1 + (2.54 + 6.52i)T \)
good5 \( 1 + (-2.36 + 1.36i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (2.52 + 1.45i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.4 + 18.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-24.7 + 14.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (17.7 - 30.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.6 + 7.86i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.1 + 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 12.4T + 961T^{2} \)
37 \( 1 + (5.80 - 10.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-26.4 - 15.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (12.6 + 21.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 84.6iT - 2.20e3T^{2} \)
53 \( 1 + (-15.1 + 8.77i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 43.1iT - 3.48e3T^{2} \)
61 \( 1 - 30.2T + 3.72e3T^{2} \)
67 \( 1 - 86.4T + 4.48e3T^{2} \)
71 \( 1 + 1.24iT - 5.04e3T^{2} \)
73 \( 1 + (6.48 + 11.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 103.T + 6.24e3T^{2} \)
83 \( 1 + (-35.2 + 20.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-15.5 - 8.95i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (2.62 + 4.54i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.07589413155348781781474643039, −12.43128465792646456373206076600, −10.78364182457262960589498068742, −9.728512063639288498247065053444, −8.238258764900669647253976416474, −7.61672114377594857700974778288, −6.30824688953223556025167027494, −5.45202268219692695414844869069, −3.38289149370778897665169657962, −0.964720409088634178473379444026, 2.38461717527742384786108737527, 3.77414171449393619629996483667, 5.19843811292744973041356675286, 6.39536196164691351729814972760, 8.567072979454376403260134000578, 9.247110914532343076704715188115, 10.21826472263989611038207368904, 11.13488221328766979094049470202, 12.08330890723483395871458385926, 13.28251171427853931157569883718

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