Properties

Label 2-224-1.1-c5-0-24
Degree $2$
Conductor $224$
Sign $-1$
Analytic cond. $35.9259$
Root an. cond. $5.99382$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.810·3-s + 71.6·5-s + 49·7-s − 242.·9-s − 569.·11-s + 137.·13-s − 58.0·15-s − 418.·17-s − 2.55e3·19-s − 39.7·21-s + 127.·23-s + 2.01e3·25-s + 393.·27-s + 2.31e3·29-s + 3.99e3·31-s + 461.·33-s + 3.51e3·35-s − 3.85e3·37-s − 111.·39-s − 4.94e3·41-s − 1.36e4·43-s − 1.73e4·45-s − 2.76e4·47-s + 2.40e3·49-s + 339.·51-s + 3.73e4·53-s − 4.08e4·55-s + ⋯
L(s)  = 1  − 0.0519·3-s + 1.28·5-s + 0.377·7-s − 0.997·9-s − 1.41·11-s + 0.225·13-s − 0.0666·15-s − 0.351·17-s − 1.62·19-s − 0.0196·21-s + 0.0500·23-s + 0.643·25-s + 0.103·27-s + 0.510·29-s + 0.746·31-s + 0.0737·33-s + 0.484·35-s − 0.462·37-s − 0.0117·39-s − 0.459·41-s − 1.12·43-s − 1.27·45-s − 1.82·47-s + 0.142·49-s + 0.0182·51-s + 1.82·53-s − 1.81·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-1$
Analytic conductor: \(35.9259\)
Root analytic conductor: \(5.99382\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 224,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 49T \)
good3 \( 1 + 0.810T + 243T^{2} \)
5 \( 1 - 71.6T + 3.12e3T^{2} \)
11 \( 1 + 569.T + 1.61e5T^{2} \)
13 \( 1 - 137.T + 3.71e5T^{2} \)
17 \( 1 + 418.T + 1.41e6T^{2} \)
19 \( 1 + 2.55e3T + 2.47e6T^{2} \)
23 \( 1 - 127.T + 6.43e6T^{2} \)
29 \( 1 - 2.31e3T + 2.05e7T^{2} \)
31 \( 1 - 3.99e3T + 2.86e7T^{2} \)
37 \( 1 + 3.85e3T + 6.93e7T^{2} \)
41 \( 1 + 4.94e3T + 1.15e8T^{2} \)
43 \( 1 + 1.36e4T + 1.47e8T^{2} \)
47 \( 1 + 2.76e4T + 2.29e8T^{2} \)
53 \( 1 - 3.73e4T + 4.18e8T^{2} \)
59 \( 1 + 3.69e4T + 7.14e8T^{2} \)
61 \( 1 + 3.80e3T + 8.44e8T^{2} \)
67 \( 1 - 2.24e4T + 1.35e9T^{2} \)
71 \( 1 + 5.50e4T + 1.80e9T^{2} \)
73 \( 1 + 6.92e4T + 2.07e9T^{2} \)
79 \( 1 + 4.09e4T + 3.07e9T^{2} \)
83 \( 1 - 1.97e4T + 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 + 9.66e4T + 8.58e9T^{2} \)
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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

−10.68314610787122361674504227114, −10.14462441451551085826847267496, −8.812306754265610636529917824803, −8.143664768615809890241917948099, −6.56583246339130667938672495269, −5.69796808371529532917433742395, −4.77377101626411512920540647055, −2.84943190522414720071472136886, −1.89248897361685020734015824899, 0, 1.89248897361685020734015824899, 2.84943190522414720071472136886, 4.77377101626411512920540647055, 5.69796808371529532917433742395, 6.56583246339130667938672495269, 8.143664768615809890241917948099, 8.812306754265610636529917824803, 10.14462441451551085826847267496, 10.68314610787122361674504227114

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