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Properties

Label 2-224-56.51-c2-0-8
Degree $2$
Conductor $224$
Sign $0.00411 + 0.999i$
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  + (−0.717 − 1.24i)3-s + (−2.27 − 1.31i)5-s + (5.39 + 4.46i)7-s + (3.47 − 6.01i)9-s + (−2.75 − 4.77i)11-s − 6.22i·13-s + 3.76i·15-s + (−10.5 − 18.2i)17-s + (12.6 − 21.8i)19-s + (1.67 − 9.89i)21-s + (−12.5 − 7.24i)23-s + (−9.05 − 15.6i)25-s − 22.8·27-s + 29.9i·29-s + (23.5 − 13.5i)31-s + ⋯
L(s)  = 1  + (−0.239 − 0.414i)3-s + (−0.454 − 0.262i)5-s + (0.770 + 0.637i)7-s + (0.385 − 0.668i)9-s + (−0.250 − 0.433i)11-s − 0.478i·13-s + 0.250i·15-s + (−0.619 − 1.07i)17-s + (0.664 − 1.15i)19-s + (0.0799 − 0.471i)21-s + (−0.545 − 0.314i)23-s + (−0.362 − 0.627i)25-s − 0.846·27-s + 1.03i·29-s + (0.758 − 0.438i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00411 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00411 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.00411 + 0.999i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.00411 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.899227 - 0.895535i\)
\(L(\frac12)\) \(\approx\) \(0.899227 - 0.895535i\)
\(L(2)\) 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 \( 1 + (-5.39 - 4.46i)T \)
good3 \( 1 + (0.717 + 1.24i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.27 + 1.31i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.75 + 4.77i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.22iT - 169T^{2} \)
17 \( 1 + (10.5 + 18.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.6 + 21.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (12.5 + 7.24i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 29.9iT - 841T^{2} \)
31 \( 1 + (-23.5 + 13.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-31.5 - 18.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 37.4T + 1.68e3T^{2} \)
43 \( 1 - 28.9T + 1.84e3T^{2} \)
47 \( 1 + (51.8 + 29.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-36.5 + 21.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-14.6 - 25.3i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (92.2 + 53.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-56.4 - 97.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 94.6iT - 5.04e3T^{2} \)
73 \( 1 + (-32.3 - 56.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-46.8 - 27.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 91.8T + 6.88e3T^{2} \)
89 \( 1 + (36.1 - 62.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 97.4T + 9.40e3T^{2} \)
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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

−11.72602424327782486762586776956, −11.20273032308777354107969476255, −9.742409480862563926257286886438, −8.731051197187260026484431339066, −7.80568674381536186522867633184, −6.73783199493196747106474064305, −5.49532211264667808691710556760, −4.39755750995367745693708303688, −2.67638798329395471783090056871, −0.75185866039655683022941711910, 1.81532487332953749779403490281, 3.89934490577362362251534907628, 4.66008220684750700748968966355, 6.03794947362539072355740535037, 7.55049866118099780788127024803, 7.973654675752256082228875407338, 9.563708297230883912540417032581, 10.49107932789557167645270153857, 11.15116466856233566711675101230, 12.08041816331417772620068339524

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