Properties

Label 2-42-7.4-c3-0-3
Degree $2$
Conductor $42$
Sign $-0.949 + 0.314i$
Analytic cond. $2.47808$
Root an. cond. $1.57419$
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 + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (−10.4 − 18.0i)5-s + 6·6-s + (−18.3 − 2.59i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−20.8 + 36.0i)10-s + (−7.58 + 13.1i)11-s + (−6.00 − 10.3i)12-s + 2.16·13-s + (13.8 + 34.3i)14-s + 62.5·15-s + (−8 − 13.8i)16-s + (59.6 − 103. i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.931 − 1.61i)5-s + 0.408·6-s + (−0.990 − 0.140i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.658 + 1.14i)10-s + (−0.207 + 0.359i)11-s + (−0.144 − 0.249i)12-s + 0.0461·13-s + (0.264 + 0.655i)14-s + 1.07·15-s + (−0.125 − 0.216i)16-s + (0.851 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $-0.949 + 0.314i$
Analytic conductor: \(2.47808\)
Root analytic conductor: \(1.57419\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :3/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0708735 - 0.439905i\)
\(L(\frac12)\) \(\approx\) \(0.0708735 - 0.439905i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (18.3 + 2.59i)T \)
good5 \( 1 + (10.4 + 18.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.58 - 13.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.16T + 2.19e3T^{2} \)
17 \( 1 + (-59.6 + 103. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.7 - 29.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.325 + 0.564i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 163.T + 2.43e4T^{2} \)
31 \( 1 + (-111. + 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (84.2 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 323.T + 6.89e4T^{2} \)
43 \( 1 - 221.T + 7.95e4T^{2} \)
47 \( 1 + (254. + 439. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-88.2 + 152. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (227. - 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (19.3 + 33.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (70.8 - 122. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 602.T + 3.57e5T^{2} \)
73 \( 1 + (-551. + 954. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-58.1 - 100. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 568.T + 5.71e5T^{2} \)
89 \( 1 + (-191. - 331. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 334.T + 9.12e5T^{2} \)
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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

−15.47340277292088298264307366055, −13.41060021625310863451117669834, −12.36125749380002901092411792638, −11.62411833615134343538942354176, −9.908676799935076287744025910980, −9.068926542399521256886882367120, −7.63038525208443911588343014653, −5.16171461966220405252643999397, −3.73622577202310628140833981999, −0.41850663403096990490243996388, 3.33872862581885107678518207610, 6.09982045050864616557324800905, 7.02002881950550454801392308947, 8.163339661400542710803417806359, 10.11224012363320507584594790054, 11.11101224817046887291525549230, 12.50229527888853378919430921389, 13.98596845067316145228808820572, 15.09341694490042941913029883420, 15.91768146279149837393198075725

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