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Properties

Label 2-224-7.6-c2-0-12
Degree $2$
Conductor $224$
Sign $-0.620 + 0.784i$
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  − 2.27i·3-s − 5.40i·5-s + (4.34 − 5.49i)7-s + 3.82·9-s − 20.9·11-s + 20.6i·13-s − 12.2·15-s − 15.2i·17-s − 30.6i·19-s + (−12.4 − 9.87i)21-s − 3.59·23-s − 4.17·25-s − 29.1i·27-s − 14.9·29-s + 23.0i·31-s + ⋯
L(s)  = 1  − 0.758i·3-s − 1.08i·5-s + (0.620 − 0.784i)7-s + 0.425·9-s − 1.90·11-s + 1.59i·13-s − 0.818·15-s − 0.898i·17-s − 1.61i·19-s + (−0.594 − 0.470i)21-s − 0.156·23-s − 0.166·25-s − 1.08i·27-s − 0.516·29-s + 0.744i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.620 + 0.784i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.620 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.603368 - 1.24654i\)
\(L(\frac12)\) \(\approx\) \(0.603368 - 1.24654i\)
\(L(2)\) 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 \( 1 + (-4.34 + 5.49i)T \)
good3 \( 1 + 2.27iT - 9T^{2} \)
5 \( 1 + 5.40iT - 25T^{2} \)
11 \( 1 + 20.9T + 121T^{2} \)
13 \( 1 - 20.6iT - 169T^{2} \)
17 \( 1 + 15.2iT - 289T^{2} \)
19 \( 1 + 30.6iT - 361T^{2} \)
23 \( 1 + 3.59T + 529T^{2} \)
29 \( 1 + 14.9T + 841T^{2} \)
31 \( 1 - 23.0iT - 961T^{2} \)
37 \( 1 - 33.5T + 1.36e3T^{2} \)
41 \( 1 + 1.85iT - 1.68e3T^{2} \)
43 \( 1 - 3.59T + 1.84e3T^{2} \)
47 \( 1 + 1.10iT - 2.20e3T^{2} \)
53 \( 1 - 56.6T + 2.80e3T^{2} \)
59 \( 1 - 74.5iT - 3.48e3T^{2} \)
61 \( 1 + 59.4iT - 3.72e3T^{2} \)
67 \( 1 - 52.7T + 4.48e3T^{2} \)
71 \( 1 - 92.5T + 5.04e3T^{2} \)
73 \( 1 - 78.2iT - 5.32e3T^{2} \)
79 \( 1 - 82.3T + 6.24e3T^{2} \)
83 \( 1 + 34.8iT - 6.88e3T^{2} \)
89 \( 1 + 65.5iT - 7.92e3T^{2} \)
97 \( 1 + 85.3iT - 9.40e3T^{2} \)
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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

−11.76140604573447311045342010956, −10.86942176055764255602727073759, −9.644222020757527887388674139034, −8.590266060203913967769501028813, −7.56171656613089958640874817821, −6.90172749620396081313654582808, −5.11317906556343203320853851085, −4.46828784800151625253809704243, −2.24023089734254622379891143565, −0.76469701371729920089800396800, 2.39862487151422565948465453111, 3.59764074531892778880235685725, 5.16866642831960016184163253058, 5.93432438967945223906294115361, 7.68976839122158187388613519810, 8.160844655582298756538199244439, 9.840134054506405640701515089637, 10.49984074487046164921422803975, 10.96621213786635884297571078173, 12.45819334338517927466618581162

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