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Properties

Label 2-224-7.5-c2-0-0
Degree $2$
Conductor $224$
Sign $-0.997 - 0.0684i$
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  + (−2.20 + 1.27i)3-s + (3.56 + 2.05i)5-s + (−6.98 + 0.407i)7-s + (−1.27 + 2.20i)9-s + (−1.63 − 2.83i)11-s − 5.88i·13-s − 10.4·15-s + (−12.0 + 6.93i)17-s + (−13.7 − 7.91i)19-s + (14.8 − 9.77i)21-s + (−18.2 + 31.6i)23-s + (−4.01 − 6.95i)25-s − 29.3i·27-s − 28.4·29-s + (−36.2 + 20.9i)31-s + ⋯
L(s)  = 1  + (−0.733 + 0.423i)3-s + (0.713 + 0.411i)5-s + (−0.998 + 0.0581i)7-s + (−0.141 + 0.244i)9-s + (−0.148 − 0.257i)11-s − 0.452i·13-s − 0.697·15-s + (−0.707 + 0.408i)17-s + (−0.721 − 0.416i)19-s + (0.707 − 0.465i)21-s + (−0.793 + 1.37i)23-s + (−0.160 − 0.278i)25-s − 1.08i·27-s − 0.981·29-s + (−1.16 + 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.997 - 0.0684i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.997 - 0.0684i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0127623 + 0.372295i\)
\(L(\frac12)\) \(\approx\) \(0.0127623 + 0.372295i\)
\(L(2)\) 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 \( 1 + (6.98 - 0.407i)T \)
good3 \( 1 + (2.20 - 1.27i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.56 - 2.05i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (1.63 + 2.83i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 5.88iT - 169T^{2} \)
17 \( 1 + (12.0 - 6.93i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.7 + 7.91i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.2 - 31.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 28.4T + 841T^{2} \)
31 \( 1 + (36.2 - 20.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.14 - 12.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 - 55.3T + 1.84e3T^{2} \)
47 \( 1 + (-29.3 - 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-42.4 - 73.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.5 + 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (25.6 + 14.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-27.4 - 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 83.8T + 5.04e3T^{2} \)
73 \( 1 + (108. - 62.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (35.1 - 60.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 27.1iT - 6.88e3T^{2} \)
89 \( 1 + (126. + 73.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 11.3iT - 9.40e3T^{2} \)
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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

−12.54938011365673858012354975323, −11.28110242650569964689458098434, −10.57089444487336427380276753774, −9.824495783817766097450362390895, −8.776642621522648082234784058281, −7.28910299154727172405806097537, −6.06562362104850041397893943086, −5.54771097453711238782354005211, −3.92766765019861120109434256628, −2.39464673530070294548699442503, 0.20429724425088123869977150741, 2.15218306056580456671119052096, 4.01522689123903629744745386687, 5.56796417033152320554905843528, 6.29579451147091094893429157359, 7.19728352329733238064851505685, 8.832862375478211798899016689916, 9.559927113595721830838655277331, 10.61552768147595045602374407417, 11.67179822385621102741769389406

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