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Properties

Label 2-224-56.5-c2-0-2
Degree $2$
Conductor $224$
Sign $-0.899 - 0.437i$
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  + (1.93 + 3.35i)3-s + (−2.33 + 4.05i)5-s + (−6.95 − 0.792i)7-s + (−2.98 + 5.17i)9-s + (−12.6 + 7.29i)11-s + 12.7·13-s − 18.1·15-s + (−16.9 + 9.76i)17-s + (8.86 − 15.3i)19-s + (−10.8 − 24.8i)21-s + (4.43 − 7.67i)23-s + (1.55 + 2.70i)25-s + 11.7·27-s + 35.4i·29-s + (−25.1 + 14.5i)31-s + ⋯
L(s)  = 1  + (0.644 + 1.11i)3-s + (−0.467 + 0.810i)5-s + (−0.993 − 0.113i)7-s + (−0.331 + 0.575i)9-s + (−1.14 + 0.663i)11-s + 0.977·13-s − 1.20·15-s + (−0.994 + 0.574i)17-s + (0.466 − 0.807i)19-s + (−0.514 − 1.18i)21-s + (0.192 − 0.333i)23-s + (0.0623 + 0.108i)25-s + 0.433·27-s + 1.22i·29-s + (−0.810 + 0.468i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.899 - 0.437i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.899 - 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.267787 + 1.16157i\)
\(L(\frac12)\) \(\approx\) \(0.267787 + 1.16157i\)
\(L(2)\) 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 \( 1 + (6.95 + 0.792i)T \)
good3 \( 1 + (-1.93 - 3.35i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.33 - 4.05i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (12.6 - 7.29i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 + (16.9 - 9.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.86 + 15.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-4.43 + 7.67i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 + (25.1 - 14.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-10.5 - 6.10i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.0iT - 1.68e3T^{2} \)
43 \( 1 - 79.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.5 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (31.3 - 18.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.20 + 2.08i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (14.6 - 25.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 + 20.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 22.6T + 5.04e3T^{2} \)
73 \( 1 + (-66.1 + 38.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-68.4 + 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 49.9T + 6.88e3T^{2} \)
89 \( 1 + (-0.970 - 0.560i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 158. iT - 9.40e3T^{2} \)
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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

−12.61835365493704609919276802072, −10.90073432474671507731560895753, −10.66366142657854883542496459246, −9.518378698938561066293667980401, −8.762957663540859513306846104971, −7.43942326992054782219443436273, −6.43923445372062730597498520301, −4.81033148420760223614593715705, −3.61335553177342246839346515185, −2.82581486268839187780223912171, 0.58901567391800494428489719855, 2.41330879917902335044275982656, 3.75029564174370930537558667557, 5.47841872355135910467811001711, 6.63874670621114155519808055549, 7.78588327266871897216383693395, 8.437490552388679893615257195845, 9.361096930188530742188216923043, 10.73056393911357458751872364214, 11.90882195042584324777578525804

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