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Properties

Label 2-224-32.21-c1-0-22
Degree $2$
Conductor $224$
Sign $-0.994 + 0.106i$
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  + (0.461 − 1.33i)2-s + (−1.66 + 0.690i)3-s + (−1.57 − 1.23i)4-s + (0.923 − 2.22i)5-s + (0.154 + 2.54i)6-s + (−0.707 − 0.707i)7-s + (−2.37 + 1.53i)8-s + (0.181 − 0.181i)9-s + (−2.55 − 2.26i)10-s + (−4.11 − 1.70i)11-s + (3.47 + 0.967i)12-s + (−0.229 − 0.554i)13-s + (−1.27 + 0.619i)14-s + 4.35i·15-s + (0.960 + 3.88i)16-s − 3.94i·17-s + ⋯
L(s)  = 1  + (0.326 − 0.945i)2-s + (−0.962 + 0.398i)3-s + (−0.787 − 0.616i)4-s + (0.412 − 0.997i)5-s + (0.0631 + 1.04i)6-s + (−0.267 − 0.267i)7-s + (−0.839 + 0.543i)8-s + (0.0605 − 0.0605i)9-s + (−0.807 − 0.715i)10-s + (−1.24 − 0.514i)11-s + (1.00 + 0.279i)12-s + (−0.0637 − 0.153i)13-s + (−0.339 + 0.165i)14-s + 1.12i·15-s + (0.240 + 0.970i)16-s − 0.955i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0354283 - 0.663153i\)
\(L(\frac12)\) \(\approx\) \(0.0354283 - 0.663153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.461 + 1.33i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.66 - 0.690i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.923 + 2.22i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (4.11 + 1.70i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.229 + 0.554i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 3.94iT - 17T^{2} \)
19 \( 1 + (1.34 + 3.25i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.34 - 3.34i)T - 23iT^{2} \)
29 \( 1 + (-3.61 + 1.49i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.67T + 31T^{2} \)
37 \( 1 + (-2.26 + 5.46i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.76 + 1.76i)T - 41iT^{2} \)
43 \( 1 + (-9.15 - 3.79i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.338iT - 47T^{2} \)
53 \( 1 + (12.0 + 5.00i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.33 + 12.8i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.69 - 1.52i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (11.6 - 4.83i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.54 - 1.54i)T + 71iT^{2} \)
73 \( 1 + (3.83 - 3.83i)T - 73iT^{2} \)
79 \( 1 - 1.21iT - 79T^{2} \)
83 \( 1 + (4.57 + 11.0i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-9.10 - 9.10i)T + 89iT^{2} \)
97 \( 1 - 0.383T + 97T^{2} \)
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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

−11.69255264174646422002609335131, −10.92177807062766917394160253165, −10.09308198561664169952911282129, −9.238809494219250644741062097912, −8.044110454728924747120948564588, −6.06407941404972849652122226600, −5.21484197896838262798246074453, −4.53118946848504692667281292375, −2.71006988970907123898950103661, −0.54498737992023922352279654587, 2.82828557897469067413130551187, 4.60850940386067041337901200961, 6.01263225574579946930292184337, 6.26948230931154809287373495236, 7.41802429628958784087675182634, 8.471958355815828328951888510679, 10.00177862821217221720372562670, 10.68974684973774924010926242840, 12.13555823144857524736979137458, 12.62078596069991967656409680440

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