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Properties

Label 2-224-7.6-c2-0-9
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

\(L(\frac{3}{2})\) \(\approx\) \(1.45771 - 0.705584i\)
\(L(\frac12)\) \(\approx\) \(1.45771 - 0.705584i\)
\(L(2)\) 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

−11.94422790746656034539365091521, −10.83693302647407307710964237437, −10.15100828635813367549982627181, −8.953568725134925612666493729128, −7.48250869868688790771254961299, −6.87653465665909169068906993957, −6.14043047740634455295378559286, −4.11882262228328505643730399959, −2.96709118756236932885191684518, −1.05816320560276776450045886826, 1.58247011576910459996181567114, 3.79588371540336047488574594390, 4.54208626374684313694785919565, 5.89593238522084077797131692499, 6.97534396006131049514095739410, 8.667551688858099694453707178901, 9.389051218452666976941570162697, 9.720393007931896085605356918947, 11.48330624062998298288964629877, 12.04969223890880409238632309791

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