Properties

Label 2-42-7.2-c1-0-1
Degree $2$
Conductor $42$
Sign $0.701 + 0.712i$
Analytic cond. $0.335371$
Root an. cond. $0.579112$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 2.59i)5-s − 0.999·6-s + (2.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (−1.5 − 2.59i)11-s + (−0.499 + 0.866i)12-s − 4·13-s + (2 − 1.73i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 1.16i)5-s − 0.408·6-s + (0.944 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (−0.452 − 0.783i)11-s + (−0.144 + 0.249i)12-s − 1.10·13-s + (0.534 − 0.462i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.335371\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738063 - 0.309310i\)
\(L(\frac12)\) \(\approx\) \(0.738063 - 0.309310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−15.60313265224890188290418727783, −14.61225741559721017060439804948, −13.65356895603610477074427325375, −12.04589492707314458522553361968, −11.38368210113026558031237446692, −10.34551278070132658000810499190, −8.258208070285795497623176922352, −6.89022930108548085633542082608, −5.05183042568136278786904085559, −2.83236694053324174912444359948, 4.39806420206531623293129256174, 5.17981846056775804588147047858, 7.43633156487582478996632849091, 8.508374544210305527887034814605, 10.08359043663864915519021896089, 11.81984739532897100902202830412, 12.56353919645869054505588797252, 14.16408823459251744051413936216, 15.21881119297981133878847469119, 16.17973659941015087962770093684

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