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Properties

Label 2-42-7.4-c5-0-4
Degree $2$
Conductor $42$
Sign $0.266 + 0.963i$
Analytic cond. $6.73612$
Root an. cond. $2.59540$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (4.5 − 7.79i)3-s + (−7.99 + 13.8i)4-s + (−43 − 74.4i)5-s + 36·6-s + (24.5 − 127. i)7-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (172 − 297. i)10-s + (−17 + 29.4i)11-s + (72 + 124. i)12-s − 3·13-s + (490 − 169. i)14-s − 774.·15-s + (−128 − 221. i)16-s + (952 − 1.64e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.769 − 1.33i)5-s + 0.408·6-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.543 − 0.942i)10-s + (−0.0423 + 0.0733i)11-s + (0.144 + 0.249i)12-s − 0.00492·13-s + (0.668 − 0.231i)14-s − 0.888·15-s + (−0.125 − 0.216i)16-s + (0.798 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(6.73612\)
Root analytic conductor: \(2.59540\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :5/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.27438 - 0.969495i\)
\(L(\frac12)\) \(\approx\) \(1.27438 - 0.969495i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 + (-24.5 + 127. i)T \)
good5 \( 1 + (43 + 74.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (17 - 29.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 3T + 3.71e5T^{2} \)
17 \( 1 + (-952 + 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-744.5 - 1.28e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-112 - 193. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 6.50e3T + 2.05e7T^{2} \)
31 \( 1 + (865.5 - 1.49e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.81e3 - 6.61e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.54e4T + 1.15e8T^{2} \)
43 \( 1 - 1.84e4T + 1.47e8T^{2} \)
47 \( 1 + (9.23e3 + 1.59e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.97e3 + 1.72e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.59e4 + 2.75e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.88e4 - 4.99e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.02e4 + 5.24e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.48e4T + 1.80e9T^{2} \)
73 \( 1 + (1.04e4 - 1.80e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.52e4 - 2.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.10e5T + 3.93e9T^{2} \)
89 \( 1 + (-2.94e4 - 5.10e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.19e5T + 8.58e9T^{2} \)
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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

−14.65670869096059589267035278023, −13.59533718097096214751554718749, −12.62707534485783479501012057098, −11.59235423336549650335378403877, −9.464709440523692675298340368460, −8.056751827785019443898501584076, −7.31543270937134885296979933300, −5.25792909075009862551866829005, −3.85567495749267176241131341764, −0.802615122734954768208848093292, 2.60770790110591299546685996876, 3.87527031054882447815796470880, 5.81909624726626437707333953063, 7.71143991980332458817599347908, 9.284111729075777432914812719963, 10.71752267874604716279027041342, 11.44387136750234889762405014090, 12.75117476496957972854954856916, 14.47216167005077790726340839807, 14.92620898687039245536545767616

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