Properties

Degree $2$
Conductor $42$
Sign $0.452 - 0.891i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.73i·3-s + 2.00·4-s + 5.91i·5-s − 2.44i·6-s + (6.24 + 3.16i)7-s − 2.82·8-s − 2.99·9-s − 8.36i·10-s − 1.75·11-s + 3.46i·12-s − 18.7i·13-s + (−8.82 − 4.47i)14-s − 10.2·15-s + 4.00·16-s − 23.4i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s + 1.18i·5-s − 0.408i·6-s + (0.891 + 0.452i)7-s − 0.353·8-s − 0.333·9-s − 0.836i·10-s − 0.159·11-s + 0.288i·12-s − 1.44i·13-s + (−0.630 − 0.319i)14-s − 0.682·15-s + 0.250·16-s − 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.452 - 0.891i$
Motivic weight: \(2\)
Character: $\chi_{42} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1),\ 0.452 - 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.715425 + 0.439281i\)
\(L(\frac12)\) \(\approx\) \(0.715425 + 0.439281i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73iT \)
7 \( 1 + (-6.24 - 3.16i)T \)
good5 \( 1 - 5.91iT - 25T^{2} \)
11 \( 1 + 1.75T + 121T^{2} \)
13 \( 1 + 18.7iT - 169T^{2} \)
17 \( 1 + 23.4iT - 289T^{2} \)
19 \( 1 - 23.0iT - 361T^{2} \)
23 \( 1 - 18.7T + 529T^{2} \)
29 \( 1 - 30T + 841T^{2} \)
31 \( 1 + 8.60iT - 961T^{2} \)
37 \( 1 + 70.9T + 1.36e3T^{2} \)
41 \( 1 + 41.3iT - 1.68e3T^{2} \)
43 \( 1 - 10.4T + 1.84e3T^{2} \)
47 \( 1 + 38.6iT - 2.20e3T^{2} \)
53 \( 1 + 37.0T + 2.80e3T^{2} \)
59 \( 1 - 97.4iT - 3.48e3T^{2} \)
61 \( 1 + 16.7iT - 3.72e3T^{2} \)
67 \( 1 - 60.9T + 4.48e3T^{2} \)
71 \( 1 + 110.T + 5.04e3T^{2} \)
73 \( 1 - 56.7iT - 5.32e3T^{2} \)
79 \( 1 + 69.8T + 6.24e3T^{2} \)
83 \( 1 + 6.43iT - 6.88e3T^{2} \)
89 \( 1 - 42.0iT - 7.92e3T^{2} \)
97 \( 1 + 51.7iT - 9.40e3T^{2} \)
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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

−15.85132384144099200782981461510, −15.05042644296659737085031902424, −14.10268182626640767903968524680, −12.04786611677819163528723866998, −10.86519446483954773602597802227, −10.13599273426427390814173032677, −8.545921331359559039305387495455, −7.27641463192859787441316648427, −5.45319462803678424647293376295, −2.93119924689143621118693881153, 1.47387573767770102315147410442, 4.76563063945721265405188761708, 6.78491589255249120465243156852, 8.240003287991511385307968377459, 9.060027996856958197971697759825, 10.83891403603784171011298473820, 11.97128720903096030219212174477, 13.13656549053271463642757229727, 14.39871840141845999389564390836, 15.90934900974497686646059791775

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