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Properties

Label 2-22-11.10-c14-0-0
Degree $2$
Conductor $22$
Sign $-0.592 + 0.805i$
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 + 2.91e3·3-s − 8.19e3·4-s − 7.53e4·5-s + 2.63e5i·6-s + 8.67e5i·7-s − 7.41e5i·8-s + 3.70e6·9-s − 6.81e6i·10-s + (−1.56e7 − 1.15e7i)11-s − 2.38e7·12-s − 3.47e7i·13-s − 7.85e7·14-s − 2.19e8·15-s + 6.71e7·16-s − 2.76e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.33·3-s − 0.500·4-s − 0.964·5-s + 0.942i·6-s + 1.05i·7-s − 0.353i·8-s + 0.774·9-s − 0.681i·10-s + (−0.805 − 0.592i)11-s − 0.666·12-s − 0.553i·13-s − 0.744·14-s − 1.28·15-s + 0.250·16-s − 0.673i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.592 + 0.805i$
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: \((2,\ 22,\ (\ :7),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.1000874033\)
\(L(\frac12)\) \(\approx\) \(0.1000874033\)
\(L(8)\) 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 \( 1 + (1.56e7 + 1.15e7i)T \)
good3 \( 1 - 2.91e3T + 4.78e6T^{2} \)
5 \( 1 + 7.53e4T + 6.10e9T^{2} \)
7 \( 1 - 8.67e5iT - 6.78e11T^{2} \)
13 \( 1 + 3.47e7iT - 3.93e15T^{2} \)
17 \( 1 + 2.76e8iT - 1.68e17T^{2} \)
19 \( 1 - 8.30e8iT - 7.99e17T^{2} \)
23 \( 1 + 5.71e9T + 1.15e19T^{2} \)
29 \( 1 + 2.61e10iT - 2.97e20T^{2} \)
31 \( 1 + 1.69e10T + 7.56e20T^{2} \)
37 \( 1 - 7.99e10T + 9.01e21T^{2} \)
41 \( 1 - 1.95e11iT - 3.79e22T^{2} \)
43 \( 1 + 2.34e11iT - 7.38e22T^{2} \)
47 \( 1 + 6.41e11T + 2.56e23T^{2} \)
53 \( 1 + 1.06e12T + 1.37e24T^{2} \)
59 \( 1 - 3.40e12T + 6.19e24T^{2} \)
61 \( 1 - 3.58e12iT - 9.87e24T^{2} \)
67 \( 1 - 4.80e11T + 3.67e25T^{2} \)
71 \( 1 - 4.31e11T + 8.27e25T^{2} \)
73 \( 1 - 4.05e12iT - 1.22e26T^{2} \)
79 \( 1 - 2.11e13iT - 3.68e26T^{2} \)
83 \( 1 - 4.92e12iT - 7.36e26T^{2} \)
89 \( 1 + 7.53e12T + 1.95e27T^{2} \)
97 \( 1 + 4.89e13T + 6.52e27T^{2} \)
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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

−15.44147302765629533056936219630, −14.50023259079482264432959214997, −13.28443044655826432489888034621, −11.82120617679612806065143899092, −9.699540810695439726515038665666, −8.224412995389234754920936208116, −7.911554316198139874005688461839, −5.72783965707668109880407744802, −3.82470337859263232371405660533, −2.51513572476022727763718295137, 0.02557120376394272374021293483, 1.91760238664358447240096101591, 3.41039496749508973876093498128, 4.36525644312441391282437206662, 7.39013932614224882418719154340, 8.342056342299151456588387763301, 9.790914296489333656298517078837, 11.10912378666026869498997419041, 12.68678954223648463106081697051, 13.80704162654583073499666855134

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