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Properties

Label 2-224-28.23-c2-0-1
Degree $2$
Conductor $224$
Sign $-0.878 + 0.477i$
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  + (−2.25 + 1.30i)3-s + (−3.59 + 6.23i)5-s + (5.36 − 4.49i)7-s + (−1.09 + 1.90i)9-s + (3.81 − 2.20i)11-s − 18.1·13-s − 18.7i·15-s + (−8.58 − 14.8i)17-s + (−19.4 − 11.2i)19-s + (−6.26 + 17.1i)21-s + (−9.53 − 5.50i)23-s + (−13.4 − 23.2i)25-s − 29.2i·27-s + 37.4·29-s + (−40.5 + 23.4i)31-s + ⋯
L(s)  = 1  + (−0.752 + 0.434i)3-s + (−0.719 + 1.24i)5-s + (0.766 − 0.641i)7-s + (−0.122 + 0.211i)9-s + (0.347 − 0.200i)11-s − 1.39·13-s − 1.25i·15-s + (−0.504 − 0.874i)17-s + (−1.02 − 0.590i)19-s + (−0.298 + 0.816i)21-s + (−0.414 − 0.239i)23-s + (−0.536 − 0.929i)25-s − 1.08i·27-s + 1.29·29-s + (−1.30 + 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.878 + 0.477i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.878 + 0.477i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0357747 - 0.140610i\)
\(L(\frac12)\) \(\approx\) \(0.0357747 - 0.140610i\)
\(L(2)\) 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 \( 1 + (-5.36 + 4.49i)T \)
good3 \( 1 + (2.25 - 1.30i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.59 - 6.23i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3.81 + 2.20i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 18.1T + 169T^{2} \)
17 \( 1 + (8.58 + 14.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (19.4 + 11.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.53 + 5.50i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 37.4T + 841T^{2} \)
31 \( 1 + (40.5 - 23.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (31.3 - 54.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 2.97T + 1.68e3T^{2} \)
43 \( 1 + 43.9iT - 1.84e3T^{2} \)
47 \( 1 + (-18.9 - 10.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-2.03 - 3.52i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-12.2 + 7.09i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.4 - 70.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.4 - 11.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 96.3iT - 5.04e3T^{2} \)
73 \( 1 + (-16.3 - 28.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (73.2 + 42.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 41.1iT - 6.88e3T^{2} \)
89 \( 1 + (49.9 - 86.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 61.8T + 9.40e3T^{2} \)
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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

−12.13221838669705906978861649354, −11.45939024749234602235620743374, −10.72680764427705353829479472761, −10.16374250756510927856124428671, −8.544558146829864719654424708040, −7.33860750953441484748260602954, −6.71598638650937371663224304576, −5.09780906652545627424742079331, −4.22858594477851228968156446276, −2.62261286639164846997530643369, 0.083600454682232193233879584370, 1.82293202511150938399695233913, 4.17217061991248299490697880555, 5.11608507561664553912309609246, 6.15421716347440983089794750071, 7.52929237787238767292429349247, 8.473980456017709981736072341633, 9.280889612878865287319954262851, 10.78377241831293811666396608784, 11.80799916693753403388776326123

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