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Properties

Label 2-224-56.3-c1-0-0
Degree $2$
Conductor $224$
Sign $0.515 - 0.856i$
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  + (0.416 + 0.240i)3-s + (1.59 + 2.76i)5-s + (−0.694 + 2.55i)7-s + (−1.38 − 2.39i)9-s + (−0.800 + 1.38i)11-s − 1.38·13-s + 1.53i·15-s + (3.48 + 2.01i)17-s + (4.56 − 2.63i)19-s + (−0.902 + 0.896i)21-s + (3.83 − 2.21i)23-s + (−2.60 + 4.50i)25-s − 2.77i·27-s − 5.10i·29-s + (0.0579 − 0.100i)31-s + ⋯
L(s)  = 1  + (0.240 + 0.138i)3-s + (0.714 + 1.23i)5-s + (−0.262 + 0.964i)7-s + (−0.461 − 0.799i)9-s + (−0.241 + 0.418i)11-s − 0.385·13-s + 0.396i·15-s + (0.845 + 0.488i)17-s + (1.04 − 0.605i)19-s + (−0.197 + 0.195i)21-s + (0.798 − 0.461i)23-s + (−0.520 + 0.901i)25-s − 0.533i·27-s − 0.948i·29-s + (0.0104 − 0.0180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.515 - 0.856i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.515 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19065 + 0.672998i\)
\(L(\frac12)\) \(\approx\) \(1.19065 + 0.672998i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.694 - 2.55i)T \)
good3 \( 1 + (-0.416 - 0.240i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.59 - 2.76i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.800 - 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.38T + 13T^{2} \)
17 \( 1 + (-3.48 - 2.01i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.56 + 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.83 + 2.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + (-0.0579 + 0.100i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.63 - 2.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.21iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (5.05 + 8.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.13 - 3.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.38 - 2.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.21 + 7.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.01 - 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (-9.30 - 5.37i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 - 5.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.87iT - 97T^{2} \)
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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

−12.28635465230168512102995291711, −11.52365037393832888658539472719, −10.25494001069912732875658149268, −9.639008705627038720556883470561, −8.649890876395014800193635318857, −7.22495516039583331855474026970, −6.27897546596540353312611272823, −5.32818908461894490952459447056, −3.32627891519694612845994125927, −2.45836339521765530787637016880, 1.29963718269918004780982539121, 3.19126986339439835497090041237, 4.91139100042197524614562990904, 5.60137472425380510716202213843, 7.24620553892124347052214939679, 8.126377925177667998780679513478, 9.216829787883910839881701734790, 10.01460698813676197019939460179, 11.07675196929060864799856913664, 12.30200275722410654662360129451

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