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Properties

Label 2-224-4.3-c2-0-0
Degree $2$
Conductor $224$
Sign $-0.707 - 0.707i$
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  − 2i·3-s − 7.29·5-s − 2.64i·7-s + 5·9-s + 14.5i·11-s − 17.8·13-s + 14.5i·15-s − 24.5·17-s + 31.1i·19-s − 5.29·21-s − 1.41i·23-s + 28.1·25-s − 28i·27-s − 21.7·29-s + 4i·31-s + ⋯
L(s)  = 1  − 0.666i·3-s − 1.45·5-s − 0.377i·7-s + 0.555·9-s + 1.32i·11-s − 1.37·13-s + 0.972i·15-s − 1.44·17-s + 1.64i·19-s − 0.251·21-s − 0.0616i·23-s + 1.12·25-s − 1.03i·27-s − 0.749·29-s + 0.129i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0698390 + 0.168606i\)
\(L(\frac12)\) \(\approx\) \(0.0698390 + 0.168606i\)
\(L(2)\) 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 \( 1 + 2.64iT \)
good3 \( 1 + 2iT - 9T^{2} \)
5 \( 1 + 7.29T + 25T^{2} \)
11 \( 1 - 14.5iT - 121T^{2} \)
13 \( 1 + 17.8T + 169T^{2} \)
17 \( 1 + 24.5T + 289T^{2} \)
19 \( 1 - 31.1iT - 361T^{2} \)
23 \( 1 + 1.41iT - 529T^{2} \)
29 \( 1 + 21.7T + 841T^{2} \)
31 \( 1 - 4iT - 961T^{2} \)
37 \( 1 - 23.4T + 1.36e3T^{2} \)
41 \( 1 + 19.4T + 1.68e3T^{2} \)
43 \( 1 + 64.9iT - 1.84e3T^{2} \)
47 \( 1 - 14.8iT - 2.20e3T^{2} \)
53 \( 1 + 91.1T + 2.80e3T^{2} \)
59 \( 1 + 24.3iT - 3.48e3T^{2} \)
61 \( 1 - 13.8T + 3.72e3T^{2} \)
67 \( 1 + 76.9iT - 4.48e3T^{2} \)
71 \( 1 - 106. iT - 5.04e3T^{2} \)
73 \( 1 - 46.6T + 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 - 1.49iT - 6.88e3T^{2} \)
89 \( 1 + 8.33T + 7.92e3T^{2} \)
97 \( 1 + 139.T + 9.40e3T^{2} \)
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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

−12.38700774070510532821866309459, −11.72117987779670467765988095357, −10.50150064962322444821008440591, −9.529297824734874998966997098916, −8.003262453710478444098831231277, −7.43404153780216395768892552895, −6.72502898921930968179892072793, −4.73997358733193454645373067448, −3.93297522565976786742545686876, −1.98890049112871225097859091013, 0.095921682444951099750097037436, 2.89104376334289199092489245187, 4.18809257159501905017591700037, 4.97894180814812614143113171911, 6.70517447710268833879562377511, 7.69951911052765232238393587128, 8.767682652436093107979081266542, 9.606390158685353695882543228015, 11.12653795645865995636741414797, 11.28211520201642836725675272246

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