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Properties

Label 2-22-11.10-c14-0-9
Degree $2$
Conductor $22$
Sign $0.874 + 0.485i$
Analytic cond. $27.3523$
Root an. cond. $5.22994$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 90.5i·2-s + 3.73e3·3-s − 8.19e3·4-s + 9.70e4·5-s − 3.37e5i·6-s + 7.69e5i·7-s + 7.41e5i·8-s + 9.13e6·9-s − 8.78e6i·10-s + (9.46e6 − 1.70e7i)11-s − 3.05e7·12-s + 4.15e7i·13-s + 6.96e7·14-s + 3.62e8·15-s + 6.71e7·16-s + 4.49e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.70·3-s − 0.500·4-s + 1.24·5-s − 1.20i·6-s + 0.934i·7-s + 0.353i·8-s + 1.91·9-s − 0.878i·10-s + (0.485 − 0.874i)11-s − 0.853·12-s + 0.662i·13-s + 0.661·14-s + 2.11·15-s + 0.250·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(27.3523\)
Root analytic conductor: \(5.22994\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :7),\ 0.874 + 0.485i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(4.314861847\)
\(L(\frac12)\) \(\approx\) \(4.314861847\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 90.5iT \)
11 \( 1 + (-9.46e6 + 1.70e7i)T \)
good3 \( 1 - 3.73e3T + 4.78e6T^{2} \)
5 \( 1 - 9.70e4T + 6.10e9T^{2} \)
7 \( 1 - 7.69e5iT - 6.78e11T^{2} \)
13 \( 1 - 4.15e7iT - 3.93e15T^{2} \)
17 \( 1 - 4.49e8iT - 1.68e17T^{2} \)
19 \( 1 - 4.22e8iT - 7.99e17T^{2} \)
23 \( 1 + 2.43e9T + 1.15e19T^{2} \)
29 \( 1 + 2.14e10iT - 2.97e20T^{2} \)
31 \( 1 - 3.99e10T + 7.56e20T^{2} \)
37 \( 1 + 6.78e10T + 9.01e21T^{2} \)
41 \( 1 + 2.81e11iT - 3.79e22T^{2} \)
43 \( 1 + 4.47e11iT - 7.38e22T^{2} \)
47 \( 1 + 4.66e9T + 2.56e23T^{2} \)
53 \( 1 + 1.15e12T + 1.37e24T^{2} \)
59 \( 1 + 3.73e12T + 6.19e24T^{2} \)
61 \( 1 - 2.74e12iT - 9.87e24T^{2} \)
67 \( 1 - 1.12e13T + 3.67e25T^{2} \)
71 \( 1 + 1.25e13T + 8.27e25T^{2} \)
73 \( 1 + 4.18e12iT - 1.22e26T^{2} \)
79 \( 1 + 2.62e13iT - 3.68e26T^{2} \)
83 \( 1 - 5.26e13iT - 7.36e26T^{2} \)
89 \( 1 + 3.77e13T + 1.95e27T^{2} \)
97 \( 1 - 8.75e13T + 6.52e27T^{2} \)
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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

−14.07612208361207512653204898379, −13.66217947679156250560259631029, −12.20173442833336267535671848512, −10.16744341187480104081004479666, −9.143576416596764242087695575957, −8.367697489745965802774071832481, −6.01130069378043499594337981020, −3.82291387221038732483082848582, −2.42712898030687205343769343696, −1.68802688611684828054128366462, 1.41393477304014995334514304937, 2.94341338633077971979140930811, 4.62099712410277578106095581648, 6.73468837556010923522710279702, 7.903627440408850093615777247553, 9.347080242436062923402019110670, 10.02745974331934258535243627801, 12.97483280095670623460356393561, 13.88294369842560255296535709784, 14.44956943577219949719437489414

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