Properties

Label 2-42-7.6-c8-0-6
Degree $2$
Conductor $42$
Sign $0.878 - 0.476i$
Analytic cond. $17.1099$
Root an. cond. $4.13641$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.3·2-s + 46.7i·3-s + 128.·4-s − 561. i·5-s + 529. i·6-s + (1.14e3 + 2.11e3i)7-s + 1.44e3·8-s − 2.18e3·9-s − 6.35e3i·10-s + 2.34e4·11-s + 5.98e3i·12-s − 2.38e4i·13-s + (1.29e4 + 2.38e4i)14-s + 2.62e4·15-s + 1.63e4·16-s + 1.28e5i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.898i·5-s + 0.408i·6-s + (0.476 + 0.878i)7-s + 0.353·8-s − 0.333·9-s − 0.635i·10-s + 1.59·11-s + 0.288i·12-s − 0.833i·13-s + (0.337 + 0.621i)14-s + 0.518·15-s + 0.250·16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.878 - 0.476i$
Analytic conductor: \(17.1099\)
Root analytic conductor: \(4.13641\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :4),\ 0.878 - 0.476i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.14475 + 0.798295i\)
\(L(\frac12)\) \(\approx\) \(3.14475 + 0.798295i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 11.3T \)
3 \( 1 - 46.7iT \)
7 \( 1 + (-1.14e3 - 2.11e3i)T \)
good5 \( 1 + 561. iT - 3.90e5T^{2} \)
11 \( 1 - 2.34e4T + 2.14e8T^{2} \)
13 \( 1 + 2.38e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.28e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.20e5iT - 1.69e10T^{2} \)
23 \( 1 - 5.01e5T + 7.83e10T^{2} \)
29 \( 1 + 6.92e5T + 5.00e11T^{2} \)
31 \( 1 + 7.48e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.40e6T + 3.51e12T^{2} \)
41 \( 1 + 9.00e5iT - 7.98e12T^{2} \)
43 \( 1 + 4.80e6T + 1.16e13T^{2} \)
47 \( 1 + 4.29e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.25e7T + 6.22e13T^{2} \)
59 \( 1 - 3.15e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.92e7iT - 1.91e14T^{2} \)
67 \( 1 - 3.58e6T + 4.06e14T^{2} \)
71 \( 1 + 3.43e7T + 6.45e14T^{2} \)
73 \( 1 + 4.52e7iT - 8.06e14T^{2} \)
79 \( 1 - 3.55e7T + 1.51e15T^{2} \)
83 \( 1 - 6.80e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.56e7iT - 3.93e15T^{2} \)
97 \( 1 - 7.87e6iT - 7.83e15T^{2} \)
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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

−14.75645486395263776296165474431, −13.02295513871772644371915749128, −12.14230502204844999347282648377, −10.99919167629095572977866292213, −9.327557964296134169846689032859, −8.267379092754792916328094762898, −6.11014374998326770570495465193, −4.96734725982819995120789546391, −3.62533851415124913919625734483, −1.51066127663975065163057466393, 1.25575346677264443652543554661, 3.06922983246473525041102474792, 4.66142258547747213658155318790, 6.75111470804738550871277285778, 7.13354045671474150652729431635, 9.235053398671610049042306239385, 11.11475125485702572323622362529, 11.61109675046216435492261539872, 13.26047281152415285852441573708, 14.21733481080611562389966350476

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