Properties

Label 2-22-11.10-c14-0-10
Degree $2$
Conductor $22$
Sign $0.148 + 0.988i$
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 − 1.88e3·3-s − 8.19e3·4-s + 1.24e5·5-s − 1.70e5i·6-s − 1.66e5i·7-s − 7.41e5i·8-s − 1.22e6·9-s + 1.12e7i·10-s + (−1.92e7 + 2.89e6i)11-s + 1.54e7·12-s + 1.04e8i·13-s + 1.50e7·14-s − 2.34e8·15-s + 6.71e7·16-s − 6.64e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.862·3-s − 0.500·4-s + 1.59·5-s − 0.609i·6-s − 0.202i·7-s − 0.353i·8-s − 0.256·9-s + 1.12i·10-s + (−0.988 + 0.148i)11-s + 0.431·12-s + 1.66i·13-s + 0.143·14-s − 1.37·15-s + 0.250·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.148 + 0.988i$
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: \((2,\ 22,\ (\ :7),\ 0.148 + 0.988i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.6342302558\)
\(L(\frac12)\) \(\approx\) \(0.6342302558\)
\(L(8)\) 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 \( 1 + (1.92e7 - 2.89e6i)T \)
good3 \( 1 + 1.88e3T + 4.78e6T^{2} \)
5 \( 1 - 1.24e5T + 6.10e9T^{2} \)
7 \( 1 + 1.66e5iT - 6.78e11T^{2} \)
13 \( 1 - 1.04e8iT - 3.93e15T^{2} \)
17 \( 1 + 6.64e8iT - 1.68e17T^{2} \)
19 \( 1 + 3.30e8iT - 7.99e17T^{2} \)
23 \( 1 + 2.67e9T + 1.15e19T^{2} \)
29 \( 1 + 2.62e10iT - 2.97e20T^{2} \)
31 \( 1 + 1.16e10T + 7.56e20T^{2} \)
37 \( 1 + 9.20e10T + 9.01e21T^{2} \)
41 \( 1 + 1.38e11iT - 3.79e22T^{2} \)
43 \( 1 + 3.30e11iT - 7.38e22T^{2} \)
47 \( 1 + 6.93e11T + 2.56e23T^{2} \)
53 \( 1 - 1.16e12T + 1.37e24T^{2} \)
59 \( 1 + 1.87e10T + 6.19e24T^{2} \)
61 \( 1 + 4.90e12iT - 9.87e24T^{2} \)
67 \( 1 + 4.98e12T + 3.67e25T^{2} \)
71 \( 1 + 1.21e13T + 8.27e25T^{2} \)
73 \( 1 - 7.76e12iT - 1.22e26T^{2} \)
79 \( 1 + 2.28e13iT - 3.68e26T^{2} \)
83 \( 1 + 1.80e13iT - 7.36e26T^{2} \)
89 \( 1 + 3.47e13T + 1.95e27T^{2} \)
97 \( 1 - 6.06e13T + 6.52e27T^{2} \)
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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

−14.10951115872002001269676126839, −13.52098504914608108004806163290, −11.76395063107132365691380793346, −10.20128974912729575931859962198, −9.082508548802380847726085863101, −7.00057487761508757096264061504, −5.86483097166831775288828338708, −4.88975627006578381448957940136, −2.18721260970537895286577726841, −0.22337133490353459945202739371, 1.44265268820598830142604279930, 2.89338038396831308292647474868, 5.35293153625864908068986426855, 5.92844085861709942171821008442, 8.433996710402742101993628329542, 10.20486715648265508971177980993, 10.67728784773291176976581483651, 12.49386244677442589927311024338, 13.27314852590461668120575752864, 14.73771456768763883733469035956

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