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Properties

Label 2-224-32.29-c1-0-20
Degree $2$
Conductor $224$
Sign $-0.555 - 0.831i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−1 − 0.414i)3-s + 2i·4-s + (−0.585 − 1.41i)5-s + (0.585 + 1.41i)6-s + (−0.707 + 0.707i)7-s + (2 − 2i)8-s + (−1.29 − 1.29i)9-s + (−0.828 + 2i)10-s + (−3.70 + 1.53i)11-s + (0.828 − 2i)12-s + (−1.41 + 3.41i)13-s + 1.41·14-s + 1.65i·15-s − 4·16-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.577 − 0.239i)3-s + i·4-s + (−0.261 − 0.632i)5-s + (0.239 + 0.577i)6-s + (−0.267 + 0.267i)7-s + (0.707 − 0.707i)8-s + (−0.430 − 0.430i)9-s + (−0.261 + 0.632i)10-s + (−1.11 + 0.462i)11-s + (0.239 − 0.577i)12-s + (−0.392 + 0.946i)13-s + 0.377·14-s + 0.427i·15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1 + 0.414i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.585 + 1.41i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (3.70 - 1.53i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.41 - 3.41i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (2.41 - 5.82i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (4.70 + 1.94i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (3.29 + 7.94i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.585 + 0.585i)T + 41iT^{2} \)
43 \( 1 + (6.12 - 2.53i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 3.17iT - 47T^{2} \)
53 \( 1 + (-9.94 + 4.12i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (5 + 12.0i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (6.82 + 2.82i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.29 - 1.36i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-4.24 + 4.24i)T - 71iT^{2} \)
73 \( 1 + (-8.82 - 8.82i)T + 73iT^{2} \)
79 \( 1 + 9.89iT - 79T^{2} \)
83 \( 1 + (5.24 - 12.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (10.8 - 10.8i)T - 89iT^{2} \)
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

−11.69882828091316296089748999134, −10.70784365669628347110865048521, −9.708922494781183330658901152423, −8.773808095506330094418033374357, −7.83958791277833236794267892183, −6.64860267540627262234622253763, −5.23259133295239489655345610239, −3.81927197185510651802505405975, −2.09573197158845129667852700435, 0, 2.82810455494575849020957114256, 4.94061853535647076738824538356, 5.74970298381492036367853470277, 6.97711291655205444164352179646, 7.81945012662596335823302185270, 8.846134160901408277104581993178, 10.24930088380581225364891352566, 10.69681994220495313839154181373, 11.44698108229398092941644236394

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