Properties

Degree $2$
Conductor $42$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s − 7-s + 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 6·13-s − 14-s + 2·15-s + 16-s + 2·17-s + 18-s − 4·19-s − 2·20-s + 21-s − 4·22-s + 8·23-s − 24-s − 25-s + 6·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{42} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868861\)
\(L(\frac12)\) \(\approx\) \(0.868861\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.81961508686075874695847513084, −15.24725628375398132189109286822, −13.48218593690234539693424068692, −12.65213238808597198203967580494, −11.37318927677395242391128273111, −10.51951977961491945205840276286, −8.356918966632091922574869118017, −6.84552480602986516155667793601, −5.33985014787602837985094112170, −3.62482887081886485478101246558, 3.62482887081886485478101246558, 5.33985014787602837985094112170, 6.84552480602986516155667793601, 8.356918966632091922574869118017, 10.51951977961491945205840276286, 11.37318927677395242391128273111, 12.65213238808597198203967580494, 13.48218593690234539693424068692, 15.24725628375398132189109286822, 15.81961508686075874695847513084

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