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Properties

Label 2-42-21.5-c1-0-0
Degree $2$
Conductor $42$
Sign $0.997 - 0.0633i$
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  + (−0.866 − 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s − 1.73i·6-s + (−2.5 − 0.866i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 + 0.866i)10-s + (−2.59 + 1.5i)11-s + (−0.866 + 1.49i)12-s − 3.46i·13-s + (1.73 + 2i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (−1.73 − 3i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.670i)5-s − 0.707i·6-s + (−0.944 − 0.327i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.474 + 0.273i)10-s + (−0.783 + 0.452i)11-s + (−0.250 + 0.433i)12-s − 0.960i·13-s + (0.462 + 0.534i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(0.335371\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671305 + 0.0212791i\)
\(L(\frac12)\) \(\approx\) \(0.671305 + 0.0212791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−16.15906084276573276161375744311, −15.29937658741090555140640662661, −13.56390779185016937782265340790, −12.69543479054155224992916983257, −10.86719071748926799172491795266, −9.828390491972964858206353755426, −9.049935931691459674212628186529, −7.53817385855293891851894222207, −5.19145891538086098469316228759, −3.12995661670097279809153507515, 2.70122917254672330594561681224, 6.13857027114088609843586671515, 7.04271017383811708247266581003, 8.556688892335385106612495292863, 9.689141903450102611835545776278, 11.16522192385471179261667390926, 12.76300415987467981604867580637, 13.79935427665260260274952683473, 14.91200113231253955780026732930, 16.09198952921829878920258598136

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