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Properties

Label 2-42-21.11-c4-0-3
Degree $2$
Conductor $42$
Sign $0.317 - 0.948i$
Analytic cond. $4.34153$
Root an. cond. $2.08363$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 + 1.41i)2-s + (3.98 + 8.06i)3-s + (3.99 − 6.92i)4-s + (35.7 − 20.6i)5-s + (−21.1 − 14.1i)6-s + (−9.50 + 48.0i)7-s + 22.6i·8-s + (−49.1 + 64.3i)9-s + (−58.3 + 101. i)10-s + (127. + 73.4i)11-s + (71.8 + 4.62i)12-s − 89.0·13-s + (−44.6 − 131. i)14-s + (309. + 205. i)15-s + (−32.0 − 55.4i)16-s + (−41.5 − 23.9i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.443 + 0.896i)3-s + (0.249 − 0.433i)4-s + (1.42 − 0.825i)5-s + (−0.588 − 0.392i)6-s + (−0.194 + 0.980i)7-s + 0.353i·8-s + (−0.606 + 0.794i)9-s + (−0.583 + 1.01i)10-s + (1.05 + 0.607i)11-s + (0.498 + 0.0321i)12-s − 0.527·13-s + (−0.227 − 0.669i)14-s + (1.37 + 0.915i)15-s + (−0.125 − 0.216i)16-s + (−0.143 − 0.0829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(4.34153\)
Root analytic conductor: \(2.08363\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :2),\ 0.317 - 0.948i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.23123 + 0.886290i\)
\(L(\frac12)\) \(\approx\) \(1.23123 + 0.886290i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 - 1.41i)T \)
3 \( 1 + (-3.98 - 8.06i)T \)
7 \( 1 + (9.50 - 48.0i)T \)
good5 \( 1 + (-35.7 + 20.6i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-127. - 73.4i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 89.0T + 2.85e4T^{2} \)
17 \( 1 + (41.5 + 23.9i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-30.9 - 53.6i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-127. + 73.7i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 1.07e3iT - 7.07e5T^{2} \)
31 \( 1 + (-524. + 909. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-1.05e3 - 1.82e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.69e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.63e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.91e3 + 1.10e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.85e3 + 1.64e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-1.28e3 - 744. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.51e3 + 2.62e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.27e3 + 3.93e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 155. iT - 2.54e7T^{2} \)
73 \( 1 + (2.31e3 - 4.01e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (2.35e3 + 4.08e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 2.60e3iT - 4.74e7T^{2} \)
89 \( 1 + (6.52e3 - 3.76e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 5.57e3T + 8.85e7T^{2} \)
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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

−15.52362190529909406694795338484, −14.60777152046739203574705291052, −13.40079627509687893698389565780, −11.83627044412763878340380197919, −9.872316364978384660746000382818, −9.487574044972330573619964040426, −8.456423416770750191259579065319, −6.18823943657509592922828889440, −4.91581172558846805390125716789, −2.14233637241364446554206873950, 1.40100332496785016752266601104, 3.07548947282370653971085177385, 6.34035774531337030504787748433, 7.20536660608586837997635527557, 8.956638383365726363374789362625, 10.03675144363081323279771611842, 11.25264405695060825454893283447, 12.83864222392289975270171148387, 13.88957141184163729064089831002, 14.52582920359855022398431103505

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