Properties

Label 2-42-7.4-c1-0-1
Degree $2$
Conductor $42$
Sign $0.605 + 0.795i$
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 + (−0.5 − 0.866i)5-s − 0.999·6-s + (0.5 + 2.59i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (−2.5 + 4.33i)11-s + (0.499 + 0.866i)12-s + (2 − 1.73i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + (−0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s + (0.188 + 0.981i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.753 + 1.30i)11-s + (0.144 + 0.249i)12-s + (0.534 − 0.462i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + (−0.117 + 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.335371\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619031 - 0.306870i\)
\(L(\frac12)\) \(\approx\) \(0.619031 - 0.306870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−15.83619965200438823876216004821, −14.80964254972782301167757611807, −13.14936952808423716407580636833, −12.41612234657133509592482402351, −11.30334526510677373324538947150, −9.631223654733318138518083191236, −8.566127481624805597415792116047, −7.23470899635064634139809584743, −4.95246024856700034661833979297, −2.46289452936338301129272717961, 3.82388470647963604033512192263, 5.81813496944578039373917664188, 7.56606074256781909532176097676, 8.576514985716478914628933745839, 10.30103747365264547233487855111, 10.93716567284204152926305098868, 13.07144868289491506618070102816, 14.28937209773509806193891437398, 15.04713662038036657454485295735, 16.48137555834193247014456911537

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