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Properties

Label 2-224-32.21-c1-0-5
Degree $2$
Conductor $224$
Sign $-0.915 - 0.401i$
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  + (0.270 + 1.38i)2-s + (0.350 − 0.145i)3-s + (−1.85 + 0.750i)4-s + (−1.27 + 3.08i)5-s + (0.296 + 0.447i)6-s + (−0.707 − 0.707i)7-s + (−1.54 − 2.37i)8-s + (−2.01 + 2.01i)9-s + (−4.62 − 0.939i)10-s + (3.31 + 1.37i)11-s + (−0.541 + 0.532i)12-s + (−1.83 − 4.43i)13-s + (0.790 − 1.17i)14-s + 1.26i·15-s + (2.87 − 2.78i)16-s + 4.85i·17-s + ⋯
L(s)  = 1  + (0.191 + 0.981i)2-s + (0.202 − 0.0838i)3-s + (−0.926 + 0.375i)4-s + (−0.571 + 1.37i)5-s + (0.120 + 0.182i)6-s + (−0.267 − 0.267i)7-s + (−0.545 − 0.838i)8-s + (−0.673 + 0.673i)9-s + (−1.46 − 0.297i)10-s + (1.00 + 0.414i)11-s + (−0.156 + 0.153i)12-s + (−0.509 − 1.22i)13-s + (0.211 − 0.313i)14-s + 0.327i·15-s + (0.718 − 0.695i)16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.915 - 0.401i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208371 + 0.994598i\)
\(L(\frac12)\) \(\approx\) \(0.208371 + 0.994598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.270 - 1.38i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.350 + 0.145i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.27 - 3.08i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-3.31 - 1.37i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.83 + 4.43i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.85iT - 17T^{2} \)
19 \( 1 + (-2.38 - 5.76i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.45 - 1.45i)T - 23iT^{2} \)
29 \( 1 + (-8.74 + 3.62i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 + (-1.67 + 4.05i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.148 + 0.148i)T - 41iT^{2} \)
43 \( 1 + (-3.45 - 1.43i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.19iT - 47T^{2} \)
53 \( 1 + (2.72 + 1.12i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.53 + 6.11i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (10.1 - 4.18i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-13.2 + 5.48i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-11.0 - 11.0i)T + 71iT^{2} \)
73 \( 1 + (9.21 - 9.21i)T - 73iT^{2} \)
79 \( 1 + 7.49iT - 79T^{2} \)
83 \( 1 + (4.15 + 10.0i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.68 + 1.68i)T + 89iT^{2} \)
97 \( 1 + 7.83T + 97T^{2} \)
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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

−12.77830838976607963061578061362, −11.85614848903249655371754776859, −10.56922039723230347319999453909, −9.811014638719882300511484925374, −8.204605482405871452469797385770, −7.71601238192910526812269066306, −6.64253920638177227325921900106, −5.73960816002249455520863089411, −4.06942421677200232846096154448, −3.03895833013272837216228078091, 0.827189829046554393676289182005, 2.88351839772614140168344985310, 4.25262183610639434388445708071, 5.05655174143832974968211457407, 6.59968530970986008019117059028, 8.431217747188791363552218073079, 9.131474037690523228280232488249, 9.525274241779540046395811569117, 11.33431972067525316957899515718, 11.94046452091278971591921627981

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