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Properties

Label 2-224-32.29-c1-0-4
Degree $2$
Conductor $224$
Sign $-0.231 - 0.972i$
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  + (−0.729 + 1.21i)2-s + (1.74 + 0.724i)3-s + (−0.934 − 1.76i)4-s + (1.44 + 3.48i)5-s + (−2.15 + 1.59i)6-s + (0.707 − 0.707i)7-s + (2.82 + 0.158i)8-s + (0.414 + 0.414i)9-s + (−5.27 − 0.794i)10-s + (−2.63 + 1.09i)11-s + (−0.354 − 3.77i)12-s + (2.43 − 5.88i)13-s + (0.340 + 1.37i)14-s + 7.14i·15-s + (−2.25 + 3.30i)16-s + 6.50i·17-s + ⋯
L(s)  = 1  + (−0.516 + 0.856i)2-s + (1.01 + 0.418i)3-s + (−0.467 − 0.884i)4-s + (0.645 + 1.55i)5-s + (−0.879 + 0.649i)6-s + (0.267 − 0.267i)7-s + (0.998 + 0.0558i)8-s + (0.138 + 0.138i)9-s + (−1.66 − 0.251i)10-s + (−0.793 + 0.328i)11-s + (−0.102 − 1.08i)12-s + (0.675 − 1.63i)13-s + (0.0910 + 0.366i)14-s + 1.84i·15-s + (−0.563 + 0.826i)16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.231 - 0.972i$
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: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828005 + 1.04861i\)
\(L(\frac12)\) \(\approx\) \(0.828005 + 1.04861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.729 - 1.21i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.74 - 0.724i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.44 - 3.48i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (2.63 - 1.09i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.43 + 5.88i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.50iT - 17T^{2} \)
19 \( 1 + (-0.581 + 1.40i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.10 + 3.10i)T + 23iT^{2} \)
29 \( 1 + (3.27 + 1.35i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 3.10T + 31T^{2} \)
37 \( 1 + (-0.403 - 0.973i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.07 - 5.07i)T + 41iT^{2} \)
43 \( 1 + (-4.48 + 1.85i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 9.40iT - 47T^{2} \)
53 \( 1 + (-3.60 + 1.49i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.33 + 8.04i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.34 - 0.972i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.931 + 0.385i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-7.12 + 7.12i)T - 71iT^{2} \)
73 \( 1 + (-3.38 - 3.38i)T + 73iT^{2} \)
79 \( 1 - 1.35iT - 79T^{2} \)
83 \( 1 + (1.64 - 3.96i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.99 - 8.99i)T - 89iT^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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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

−13.03888619794121314167845906355, −10.85198271605115026591054374013, −10.40316636500096959050962459707, −9.693084617958360211734424678054, −8.301614444851111640227623144633, −7.82138392026027666226965547400, −6.50280495286294592007553891405, −5.62515713278142412201596345592, −3.76653987366329402438654731739, −2.42690411936870669454549707779, 1.46706220249815093055100726702, 2.55317928123725888704357227897, 4.25935273566183382557850934936, 5.47352533190448181672612680028, 7.49240870228170073765845449699, 8.382650999600741062549546707650, 9.134450373371759772474725326219, 9.528246737958223029995786827614, 11.13868441494947039892059546891, 12.03609844004994373136150444102

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