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Properties

Label 2-224-7.6-c2-0-11
Degree $2$
Conductor $224$
Sign $0.483 + 0.875i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56i·3-s − 3.23i·5-s + (−3.38 − 6.12i)7-s + 6.56·9-s − 3.93·11-s − 14.1i·13-s + 5.04·15-s − 26.1i·17-s + 10.6i·19-s + (9.56 − 5.27i)21-s + 20.9·23-s + 14.5·25-s + 24.2i·27-s + 13.1·29-s − 39.8i·31-s + ⋯
L(s)  = 1  + 0.520i·3-s − 0.646i·5-s + (−0.483 − 0.875i)7-s + 0.729·9-s − 0.357·11-s − 1.08i·13-s + 0.336·15-s − 1.53i·17-s + 0.563i·19-s + (0.455 − 0.251i)21-s + 0.908·23-s + 0.582·25-s + 0.899i·27-s + 0.452·29-s − 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.483 + 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20567 - 0.711792i\)
\(L(\frac12)\) \(\approx\) \(1.20567 - 0.711792i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (3.38 + 6.12i)T \)
good3 \( 1 - 1.56iT - 9T^{2} \)
5 \( 1 + 3.23iT - 25T^{2} \)
11 \( 1 + 3.93T + 121T^{2} \)
13 \( 1 + 14.1iT - 169T^{2} \)
17 \( 1 + 26.1iT - 289T^{2} \)
19 \( 1 - 10.6iT - 361T^{2} \)
23 \( 1 - 20.9T + 529T^{2} \)
29 \( 1 - 13.1T + 841T^{2} \)
31 \( 1 + 39.8iT - 961T^{2} \)
37 \( 1 + 1.12T + 1.36e3T^{2} \)
41 \( 1 + 2.36iT - 1.68e3T^{2} \)
43 \( 1 + 59.2T + 1.84e3T^{2} \)
47 \( 1 + 33.6iT - 2.20e3T^{2} \)
53 \( 1 + 30T + 2.80e3T^{2} \)
59 \( 1 - 63.3iT - 3.48e3T^{2} \)
61 \( 1 - 103. iT - 3.72e3T^{2} \)
67 \( 1 - 28.7T + 4.48e3T^{2} \)
71 \( 1 + 90.3T + 5.04e3T^{2} \)
73 \( 1 - 62.5iT - 5.32e3T^{2} \)
79 \( 1 - 139.T + 6.24e3T^{2} \)
83 \( 1 + 109. iT - 6.88e3T^{2} \)
89 \( 1 + 2.04iT - 7.92e3T^{2} \)
97 \( 1 - 90.7iT - 9.40e3T^{2} \)
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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

−11.89851750523055373190657754674, −10.62394892063625890993268539665, −10.00654590292405131606466177528, −9.110839788763895075007527611004, −7.81081048241329319148267062507, −6.91737312476640506574896628292, −5.34000673346758254706633579584, −4.43331353546425010104741425265, −3.12785674896288581037385834987, −0.810174228328889964159929270673, 1.83236468482907489337941901580, 3.23729871471545297064946763680, 4.83810840398826483059235908956, 6.41579196632958143902589867031, 6.85466843154247242758353482055, 8.222586254089876051436506482972, 9.232680676636698382145695607452, 10.31351119365763498190438935323, 11.23730365000488628626335796965, 12.41451788011349638651098069419

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