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Properties

Label 2-224-56.53-c1-0-1
Degree $2$
Conductor $224$
Sign $0.907 - 0.419i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.787i)3-s + (−0.476 + 0.274i)5-s + (2.60 − 0.447i)7-s + (−0.258 − 0.447i)9-s + (2.07 + 1.19i)11-s + 3.96i·13-s − 0.866·15-s + (2.10 − 3.65i)17-s + (−5.75 + 3.32i)19-s + (3.91 + 1.44i)21-s + (−1.17 − 2.03i)23-s + (−2.34 + 4.06i)25-s − 5.54i·27-s − 8.21i·29-s + (0.433 − 0.750i)31-s + ⋯
L(s)  = 1  + (0.787 + 0.454i)3-s + (−0.212 + 0.122i)5-s + (0.985 − 0.169i)7-s + (−0.0862 − 0.149i)9-s + (0.624 + 0.360i)11-s + 1.10i·13-s − 0.223·15-s + (0.511 − 0.885i)17-s + (−1.31 + 0.761i)19-s + (0.853 + 0.314i)21-s + (−0.244 − 0.424i)23-s + (−0.469 + 0.813i)25-s − 1.06i·27-s − 1.52i·29-s + (0.0777 − 0.134i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55048 + 0.341362i\)
\(L(\frac12)\) \(\approx\) \(1.55048 + 0.341362i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 0.447i)T \)
good3 \( 1 + (-1.36 - 0.787i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.476 - 0.274i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.07 - 1.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.96iT - 13T^{2} \)
17 \( 1 + (-2.10 + 3.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.75 - 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 + 2.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.21iT - 29T^{2} \)
31 \( 1 + (-0.433 + 0.750i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.229 - 0.132i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.35iT - 43T^{2} \)
47 \( 1 + (-1.29 - 2.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.36 + 5.40i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.26 - 1.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.18 - 3.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.31 + 1.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (2.33 - 4.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.308 - 0.533i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 + (-3.19 - 5.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−12.01138282828078247800720346712, −11.54279684797396385348512711949, −10.24104044858675634562639687594, −9.311027143601636137185495426460, −8.469415078479252204783559511560, −7.50583915650002888710542589301, −6.23851096096979312769040567294, −4.57936810101933339203002193650, −3.76261424563357664831361057640, −2.04771490383933926022573151910, 1.73344062487906927084893924292, 3.23544793753380519967675762542, 4.72729817148535302420936823678, 6.02814029650708959733935128590, 7.47334721359317041824071624711, 8.325400633724263233422213680699, 8.793249529956974795369299341543, 10.39460575736507845217280785752, 11.19504149655606952425897029835, 12.33457622768147779744702576559

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