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Properties

Label 2-224-32.29-c1-0-2
Degree $2$
Conductor $224$
Sign $-0.0203 - 0.999i$
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  + (−1.32 + 0.494i)2-s + (0.0422 + 0.0175i)3-s + (1.51 − 1.31i)4-s + (1.39 + 3.36i)5-s + (−0.0646 − 0.00227i)6-s + (−0.707 + 0.707i)7-s + (−1.35 + 2.48i)8-s + (−2.11 − 2.11i)9-s + (−3.51 − 3.76i)10-s + (4.63 − 1.91i)11-s + (0.0868 − 0.0289i)12-s + (−2.45 + 5.92i)13-s + (0.586 − 1.28i)14-s + 0.166i·15-s + (0.561 − 3.96i)16-s + 1.91i·17-s + ⋯
L(s)  = 1  + (−0.936 + 0.349i)2-s + (0.0244 + 0.0101i)3-s + (0.755 − 0.655i)4-s + (0.623 + 1.50i)5-s + (−0.0264 − 0.000929i)6-s + (−0.267 + 0.267i)7-s + (−0.477 + 0.878i)8-s + (−0.706 − 0.706i)9-s + (−1.11 − 1.19i)10-s + (1.39 − 0.578i)11-s + (0.0250 − 0.00836i)12-s + (−0.680 + 1.64i)13-s + (0.156 − 0.343i)14-s + 0.0430i·15-s + (0.140 − 0.990i)16-s + 0.463i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0203 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0203 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.0203 - 0.999i$
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: \((2,\ 224,\ (\ :1/2),\ -0.0203 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586102 + 0.598153i\)
\(L(\frac12)\) \(\approx\) \(0.586102 + 0.598153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.494i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.0422 - 0.0175i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.39 - 3.36i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (-4.63 + 1.91i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.45 - 5.92i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 1.91iT - 17T^{2} \)
19 \( 1 + (0.815 - 1.96i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.13 - 3.13i)T + 23iT^{2} \)
29 \( 1 + (-1.40 - 0.582i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.47T + 31T^{2} \)
37 \( 1 + (2.60 + 6.29i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.49 - 2.49i)T + 41iT^{2} \)
43 \( 1 + (-9.16 + 3.79i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.30iT - 47T^{2} \)
53 \( 1 + (-7.05 + 2.92i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.45 + 8.34i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.81 - 0.750i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (7.17 + 2.97i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-0.959 + 0.959i)T - 71iT^{2} \)
73 \( 1 + (5.23 + 5.23i)T + 73iT^{2} \)
79 \( 1 + 5.76iT - 79T^{2} \)
83 \( 1 + (-5.90 + 14.2i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-0.788 + 0.788i)T - 89iT^{2} \)
97 \( 1 - 9.66T + 97T^{2} \)
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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

−11.98634163317794960276296194439, −11.40036529670306407057664009287, −10.45800034114314520285796947025, −9.311984859928743433223223836689, −8.985894612395354657373278902483, −7.24208350412437742105139691515, −6.50493976222878097010308993348, −5.91545046323954647793162887582, −3.49345360749646893032056795127, −2.07009730702667306355959484761, 0.986395198176511521521373807708, 2.66105068786979806444700790325, 4.56828530466755658120949473590, 5.81503311222280295432772812642, 7.24238743060564224964047072996, 8.361580213760692857757266557823, 9.107887288178811473635268184403, 9.863757620167130025426210555576, 10.87739377298326670282986447097, 12.11648757141957297324725311065

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