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Properties

Label 2-42-3.2-c6-0-7
Degree $2$
Conductor $42$
Sign $-0.0392 + 0.999i$
Analytic cond. $9.66227$
Root an. cond. $3.10841$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s + (−26.9 − 1.06i)3-s − 32.0·4-s + 222. i·5-s + (6.00 − 152. i)6-s − 129.·7-s − 181. i·8-s + (726. + 57.2i)9-s − 1.25e3·10-s − 2.29e3i·11-s + (863. + 33.9i)12-s − 3.12e3·13-s − 733. i·14-s + (235. − 5.99e3i)15-s + 1.02e3·16-s − 695. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.999 − 0.0392i)3-s − 0.500·4-s + 1.77i·5-s + (0.0277 − 0.706i)6-s − 0.377·7-s − 0.353i·8-s + (0.996 + 0.0785i)9-s − 1.25·10-s − 1.72i·11-s + (0.499 + 0.0196i)12-s − 1.42·13-s − 0.267i·14-s + (0.0698 − 1.77i)15-s + 0.250·16-s − 0.141i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0392 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0392 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $-0.0392 + 0.999i$
Analytic conductor: \(9.66227\)
Root analytic conductor: \(3.10841\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :3),\ -0.0392 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0141287 - 0.0146952i\)
\(L(\frac12)\) \(\approx\) \(0.0141287 - 0.0146952i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
3 \( 1 + (26.9 + 1.06i)T \)
7 \( 1 + 129.T \)
good5 \( 1 - 222. iT - 1.56e4T^{2} \)
11 \( 1 + 2.29e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.12e3T + 4.82e6T^{2} \)
17 \( 1 + 695. iT - 2.41e7T^{2} \)
19 \( 1 - 9.06e3T + 4.70e7T^{2} \)
23 \( 1 - 202. iT - 1.48e8T^{2} \)
29 \( 1 - 2.08e3iT - 5.94e8T^{2} \)
31 \( 1 + 1.65e4T + 8.87e8T^{2} \)
37 \( 1 + 2.22e4T + 2.56e9T^{2} \)
41 \( 1 + 4.61e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.07e5T + 6.32e9T^{2} \)
47 \( 1 - 9.75e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.81e5iT - 2.21e10T^{2} \)
59 \( 1 + 7.86e4iT - 4.21e10T^{2} \)
61 \( 1 + 2.80e5T + 5.15e10T^{2} \)
67 \( 1 + 3.21e5T + 9.04e10T^{2} \)
71 \( 1 - 1.78e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.03e5T + 1.51e11T^{2} \)
79 \( 1 + 1.38e5T + 2.43e11T^{2} \)
83 \( 1 + 2.83e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.80e5iT - 4.96e11T^{2} \)
97 \( 1 + 2.97e5T + 8.32e11T^{2} \)
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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

−14.54300574450443601983702462928, −13.58303785838034379859298007354, −11.86426621010350305100228221977, −10.82318275345804182296572936528, −9.746300141329960665450776883279, −7.52241868050734990152932565804, −6.60940257905672799850854718588, −5.49858132430867758280345709905, −3.25670251118751863733370066813, −0.01121156598889350648599071023, 1.55423184794616157201628859418, 4.50641642732292627442299255938, 5.24686507994265154514176939866, 7.43168985245629159780682016054, 9.381465569975047278208864181548, 10.01860472137865242073967808894, 12.01771158804892559849824740167, 12.29089511130576262922980318991, 13.27158945569123431757503702521, 15.25272754860025824749687706115

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