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Properties

Label 2-42-21.17-c1-0-1
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: $\chi_{42} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.853264 - 0.0270469i\)
\(L(\frac12)\) \(\approx\) \(0.853264 - 0.0270469i\)
\(L(\frac{3}{2})\) 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

−16.12595500116077406702952094491, −15.03987385017330370179528767840, −13.71914052363018378895793882181, −12.13473529783394500416242962456, −11.72797026111171284754358909540, −9.960982919127179218154463817591, −9.105432201655177709016374067598, −6.62568232422004867577611727202, −5.07405313003115312716722025638, −3.70646706726554967046764789054, 3.42734786389487509346502907389, 5.84947314977811634879342655703, 6.84638296495950920685944361414, 8.064618417518669676248232243541, 10.32431591829943517042948545711, 11.65137185427555749833535998976, 12.65813514585804913359573069201, 13.69091443436195335528792289215, 14.79226336569032848716236724080, 16.16476878162837239238682102134

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