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Properties

Label 2-224-7.4-c1-0-7
Degree $2$
Conductor $224$
Sign $-0.198 + 0.980i$
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  + (1.20 − 2.09i)3-s + (−1.91 − 3.31i)5-s + (1 + 2.44i)7-s + (−1.41 − 2.44i)9-s + (0.207 − 0.358i)11-s − 2.82·13-s − 9.24·15-s + (2.91 − 5.04i)17-s + (1.79 + 3.10i)19-s + (6.32 + 0.866i)21-s + (1.62 + 2.80i)23-s + (−4.82 + 8.36i)25-s + 0.414·27-s + 2.82·29-s + (4.20 − 7.28i)31-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.856 − 1.48i)5-s + (0.377 + 0.925i)7-s + (−0.471 − 0.816i)9-s + (0.0624 − 0.108i)11-s − 0.784·13-s − 2.38·15-s + (0.706 − 1.22i)17-s + (0.411 + 0.712i)19-s + (1.38 + 0.188i)21-s + (0.338 + 0.585i)23-s + (−0.965 + 1.67i)25-s + 0.0797·27-s + 0.525·29-s + (0.755 − 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844766 - 1.03253i\)
\(L(\frac12)\) \(\approx\) \(0.844766 - 1.03253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2.44i)T \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.91 + 3.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.207 + 0.358i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (-2.91 + 5.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.79 - 3.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.62 - 2.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-4.20 + 7.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.32 - 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.17T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 + (-3.79 - 6.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.44 - 7.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.32 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 + (-1.67 + 2.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.03 + 6.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.82T + 97T^{2} \)
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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

−11.98389527962239673048910508638, −11.76013992815798216651585201982, −9.617757617717251643877824008287, −8.774810531444504115688774683425, −7.942184246966596231017470418944, −7.41484753012266688206776437691, −5.67397719754822427760497833740, −4.59673674289179745639317948363, −2.77961921529794564951490862901, −1.18799524278326479230258955465, 2.91798100499612039371382808529, 3.78261020861227705606017715149, 4.73721283945033151759484524394, 6.70459373576304116152620958514, 7.57814932740326661071945422621, 8.550211721877067638018216529020, 9.991091123629378655314275699201, 10.44580801002007241902780874961, 11.18897976348401153406923243078, 12.36276636613793200641842647485

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