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Properties

Label 2-224-224.83-c1-0-9
Degree $2$
Conductor $224$
Sign $-0.345 - 0.938i$
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  + (0.581 + 1.28i)2-s + (−1.32 + 1.50i)4-s + (1.87 + 1.87i)7-s + (−2.70 − 0.832i)8-s + (2.12 + 2.12i)9-s + (−3.10 + 1.28i)11-s + (−1.32 + 3.5i)14-s + (−0.5 − 3.96i)16-s + (−1.5 + 3.96i)18-s + (−3.46 − 3.25i)22-s + (1.35 − 1.35i)23-s + (3.53 − 3.53i)25-s + (−5.28 + 0.331i)28-s + (2.25 − 5.44i)29-s + (4.82 − 2.95i)32-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.661 + 0.750i)4-s + (0.707 + 0.707i)7-s + (−0.955 − 0.294i)8-s + (0.707 + 0.707i)9-s + (−0.935 + 0.387i)11-s + (−0.353 + 0.935i)14-s + (−0.125 − 0.992i)16-s + (−0.353 + 0.935i)18-s + (−0.737 − 0.693i)22-s + (0.282 − 0.282i)23-s + (0.707 − 0.707i)25-s + (−0.998 + 0.0626i)28-s + (0.419 − 1.01i)29-s + (0.852 − 0.522i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818137 + 1.17333i\)
\(L(\frac12)\) \(\approx\) \(0.818137 + 1.17333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.581 - 1.28i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good3 \( 1 + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (3.10 - 1.28i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.35 + 1.35i)T - 23iT^{2} \)
29 \( 1 + (-2.25 + 5.44i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-11.1 + 4.62i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 - 41iT^{2} \)
43 \( 1 + (8.37 - 3.46i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (2.27 + 5.49i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.19 + 0.910i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (11.3 + 11.3i)T + 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 + (58.6 + 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.79806619845785565192788799369, −11.83112169856387863357910859356, −10.60119068855547927246846613832, −9.447189178622155688085173857641, −8.204593251166645939510210520578, −7.68361920601742971333538693282, −6.39551871434317278641608072845, −5.16042019895046236874709084947, −4.46779481631165699772300407475, −2.52951006492490064765837497611, 1.24021335724224941668352710528, 3.09001856668160577262137186420, 4.34575485342817399004896867834, 5.33890828626082011841436534888, 6.81066829875610638799238007046, 8.101037354027003923864086913790, 9.285694425682301023815070403537, 10.30976033260929922613915616111, 10.97399070096593634936879092780, 11.92006896772254737502753452704

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