Properties

Degree $2$
Conductor $42$
Sign $0.947 - 0.320i$
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 + 1.01i·5-s + 2.44i·6-s + (−2.24 − 6.63i)7-s + 2.82·8-s − 2.99·9-s + 1.43i·10-s − 10.2·11-s + 3.46i·12-s − 8.95i·13-s + (−3.17 − 9.37i)14-s − 1.75·15-s + 4.00·16-s + 30.4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.202i·5-s + 0.408i·6-s + (−0.320 − 0.947i)7-s + 0.353·8-s − 0.333·9-s + 0.143i·10-s − 0.931·11-s + 0.288i·12-s − 0.689i·13-s + (−0.226 − 0.669i)14-s − 0.117·15-s + 0.250·16-s + 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.947 - 0.320i$
Motivic weight: \(2\)
Character: $\chi_{42} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :1),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44507 + 0.237750i\)
\(L(\frac12)\) \(\approx\) \(1.44507 + 0.237750i\)
\(L(2)\) 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 \( 1 + (2.24 + 6.63i)T \)
good5 \( 1 - 1.01iT - 25T^{2} \)
11 \( 1 + 10.2T + 121T^{2} \)
13 \( 1 + 8.95iT - 169T^{2} \)
17 \( 1 - 30.4iT - 289T^{2} \)
19 \( 1 + 16.1iT - 361T^{2} \)
23 \( 1 + 6.72T + 529T^{2} \)
29 \( 1 - 30T + 841T^{2} \)
31 \( 1 - 50.1iT - 961T^{2} \)
37 \( 1 - 30.9T + 1.36e3T^{2} \)
41 \( 1 + 7.10iT - 1.68e3T^{2} \)
43 \( 1 + 74.4T + 1.84e3T^{2} \)
47 \( 1 + 58.2iT - 2.20e3T^{2} \)
53 \( 1 + 70.9T + 2.80e3T^{2} \)
59 \( 1 + 0.492iT - 3.48e3T^{2} \)
61 \( 1 - 2.86iT - 3.72e3T^{2} \)
67 \( 1 - 27.0T + 4.48e3T^{2} \)
71 \( 1 - 50.6T + 5.04e3T^{2} \)
73 \( 1 + 70.6iT - 5.32e3T^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 + 104. iT - 6.88e3T^{2} \)
89 \( 1 - 144. iT - 7.92e3T^{2} \)
97 \( 1 + 100. iT - 9.40e3T^{2} \)
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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

−15.68039694916349266716743269178, −14.76786684987461435128296966158, −13.52483304984713187854458615402, −12.58757851877157377035152372748, −10.80692096612039851866663777859, −10.26337553007293711497230863901, −8.203509634533493905314742065359, −6.58411108149393790950686580729, −4.90431479939002232601911369007, −3.31428425515721586056396944495, 2.65424106596318642352946289616, 5.05233619897191696013507298883, 6.44305515098406358521201467802, 7.975142494930554989947163677204, 9.577375766818075752095347936206, 11.41357024915097376533190243079, 12.33427126077361109316254439450, 13.33005525138254894059971095074, 14.41625473725095572157060235075, 15.71171946013188735139765502039

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