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Properties

Label 2-126-63.5-c3-0-0
Degree $2$
Conductor $126$
Sign $-0.0747 + 0.997i$
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  + 2i·2-s + (−2.75 + 4.40i)3-s − 4·4-s + (1.48 + 2.56i)5-s + (−8.81 − 5.50i)6-s + (−18.4 − 0.970i)7-s − 8i·8-s + (−11.8 − 24.2i)9-s + (−5.12 + 2.96i)10-s + (−30.3 − 17.5i)11-s + (11.0 − 17.6i)12-s + (73.4 + 42.3i)13-s + (1.94 − 36.9i)14-s + (−15.3 − 0.539i)15-s + 16·16-s + (−20.8 − 36.1i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.530 + 0.847i)3-s − 0.5·4-s + (0.132 + 0.229i)5-s + (−0.599 − 0.374i)6-s + (−0.998 − 0.0524i)7-s − 0.353i·8-s + (−0.438 − 0.898i)9-s + (−0.162 + 0.0936i)10-s + (−0.831 − 0.479i)11-s + (0.265 − 0.423i)12-s + (1.56 + 0.904i)13-s + (0.0370 − 0.706i)14-s + (−0.264 − 0.00928i)15-s + 0.250·16-s + (−0.297 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0747 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0747 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0747 + 0.997i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.0747 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0256958 - 0.0276939i\)
\(L(\frac12)\) \(\approx\) \(0.0256958 - 0.0276939i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 + (2.75 - 4.40i)T \)
7 \( 1 + (18.4 + 0.970i)T \)
good5 \( 1 + (-1.48 - 2.56i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (30.3 + 17.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-73.4 - 42.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (20.8 + 36.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (112. + 64.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-35.6 + 20.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (258. - 149. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 191. iT - 2.97e4T^{2} \)
37 \( 1 + (181. - 313. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (105. - 182. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (19.8 + 34.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 464.T + 1.03e5T^{2} \)
53 \( 1 + (339. - 195. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 524.T + 2.05e5T^{2} \)
61 \( 1 + 75.6iT - 2.26e5T^{2} \)
67 \( 1 + 267.T + 3.00e5T^{2} \)
71 \( 1 - 493. iT - 3.57e5T^{2} \)
73 \( 1 + (-64.5 + 37.2i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 809.T + 4.93e5T^{2} \)
83 \( 1 + (-17.7 - 30.6i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-640. + 1.10e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-246. + 142. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.62988100778184140216371571410, −12.91316626358110872211606050687, −11.31032788215041510906856155846, −10.55735814805137934460304101156, −9.357727286257508973825858862973, −8.563590562594735854701248301530, −6.68929904607865705917128307755, −6.09453969068095882049629894038, −4.66486254953243387860438384115, −3.33408859055048735350806040100, 0.02024202017741314103512008865, 1.81085218993937129574351795278, 3.51253545030578752298475630487, 5.41154806705745888627596952175, 6.38512516022647128048554131330, 7.87003368257559228378942085071, 8.960099061481758456042651228737, 10.45498237790275618931892171680, 10.97553837932293380664957207933, 12.50070948113956571583902925701

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