Properties

Label 2-42-7.5-c8-0-9
Degree $2$
Conductor $42$
Sign $-0.727 + 0.686i$
Analytic cond. $17.1099$
Root an. cond. $4.13641$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 − 9.79i)2-s + (40.5 − 23.3i)3-s + (−63.9 − 110. i)4-s + (−65.1 − 37.6i)5-s − 529. i·6-s + (2.24e3 − 843. i)7-s − 1.44e3·8-s + (1.09e3 − 1.89e3i)9-s + (−737. + 425. i)10-s + (−8.03e3 − 1.39e4i)11-s + (−5.18e3 − 2.99e3i)12-s − 1.63e4i·13-s + (4.44e3 − 2.67e4i)14-s − 3.52e3·15-s + (−8.19e3 + 1.41e4i)16-s + (−6.12e3 + 3.53e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.104 − 0.0602i)5-s − 0.408i·6-s + (0.936 − 0.351i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0737 + 0.0425i)10-s + (−0.548 − 0.950i)11-s + (−0.249 − 0.144i)12-s − 0.572i·13-s + (0.115 − 0.697i)14-s − 0.0695·15-s + (−0.125 + 0.216i)16-s + (−0.0733 + 0.0423i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $-0.727 + 0.686i$
Analytic conductor: \(17.1099\)
Root analytic conductor: \(4.13641\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :4),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.873888 - 2.20067i\)
\(L(\frac12)\) \(\approx\) \(0.873888 - 2.20067i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 + 9.79i)T \)
3 \( 1 + (-40.5 + 23.3i)T \)
7 \( 1 + (-2.24e3 + 843. i)T \)
good5 \( 1 + (65.1 + 37.6i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (8.03e3 + 1.39e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 1.63e4iT - 8.15e8T^{2} \)
17 \( 1 + (6.12e3 - 3.53e3i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (3.59e4 + 2.07e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.52e4 - 2.64e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 4.96e5T + 5.00e11T^{2} \)
31 \( 1 + (2.33e5 - 1.34e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-7.46e5 + 1.29e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 3.80e5iT - 7.98e12T^{2} \)
43 \( 1 - 3.70e6T + 1.16e13T^{2} \)
47 \( 1 + (-6.05e6 - 3.49e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-2.69e6 - 4.67e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-5.16e6 + 2.98e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-2.18e7 - 1.26e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.57e7 - 2.73e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 2.18e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.52e7 + 1.45e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-1.26e7 + 2.19e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 2.26e7iT - 2.25e15T^{2} \)
89 \( 1 + (1.00e8 + 5.79e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 3.41e7iT - 7.83e15T^{2} \)
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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

−13.76391128911765091301673417646, −12.75725416679579006150824386486, −11.40473355511212214340996658339, −10.42372498254558904277650989692, −8.755886224305763245624070821718, −7.64321459399796406444116096081, −5.64466150086337152483465208269, −4.02386703040634880841378071536, −2.42747229743590644103152421069, −0.77236595191753250409894995144, 2.14748705475370895761836519408, 4.11299602629671579622778240645, 5.34371069149206671315904727901, 7.21401884384241471738971683744, 8.292082052158277893666429080157, 9.584171798389084390083798504417, 11.20415316722355259160132932503, 12.55272975145228598022939420714, 13.82395793014186460767590700940, 14.87601515668325943749531482708

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