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Properties

Label 2-224-56.45-c2-0-5
Degree $2$
Conductor $224$
Sign $0.596 - 0.802i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 + 2.02i)3-s + (−1.55 − 2.68i)5-s + (6.89 − 1.21i)7-s + (1.77 + 3.06i)9-s + (4.06 + 2.34i)11-s − 6.88·13-s + 7.24·15-s + (14.7 + 8.49i)17-s + (13.1 + 22.7i)19-s + (−5.59 + 15.3i)21-s + (12.9 + 22.4i)23-s + (7.69 − 13.3i)25-s − 29.3·27-s + 42.2i·29-s + (−15.9 − 9.18i)31-s + ⋯
L(s)  = 1  + (−0.389 + 0.674i)3-s + (−0.310 − 0.537i)5-s + (0.984 − 0.173i)7-s + (0.196 + 0.341i)9-s + (0.369 + 0.213i)11-s − 0.529·13-s + 0.482·15-s + (0.865 + 0.499i)17-s + (0.689 + 1.19i)19-s + (−0.266 + 0.731i)21-s + (0.562 + 0.974i)23-s + (0.307 − 0.532i)25-s − 1.08·27-s + 1.45i·29-s + (−0.512 − 0.296i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30046 + 0.653330i\)
\(L(\frac12)\) \(\approx\) \(1.30046 + 0.653330i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-6.89 + 1.21i)T \)
good3 \( 1 + (1.16 - 2.02i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.55 + 2.68i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-4.06 - 2.34i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 6.88T + 169T^{2} \)
17 \( 1 + (-14.7 - 8.49i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.1 - 22.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.9 - 22.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 42.2iT - 841T^{2} \)
31 \( 1 + (15.9 + 9.18i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-43.1 + 24.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 10.7iT - 1.68e3T^{2} \)
43 \( 1 + 24.1iT - 1.84e3T^{2} \)
47 \( 1 + (11.8 - 6.84i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-6.03 - 3.48i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-53.0 + 91.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.7 + 80.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-77.2 - 44.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 81.7T + 5.04e3T^{2} \)
73 \( 1 + (119. + 68.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (6.55 + 11.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 2.15T + 6.88e3T^{2} \)
89 \( 1 + (87.8 - 50.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 88.9iT - 9.40e3T^{2} \)
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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

−12.04504155815187526527220732936, −11.18778281680389571155916954066, −10.29239196976201580880693257638, −9.383873984858754794110138366378, −8.115690733099672635479460133185, −7.37716853589239911901018856354, −5.57359534782317059002933146279, −4.83144702727735254517453230953, −3.75834638699405293624454759112, −1.50390832425996056557446751051, 0.997564513355930486717655762098, 2.80282017051283148368493757210, 4.48449034753967905609073294551, 5.74186198426185260089316356802, 7.00432469480336632667826872319, 7.57584539586087678130332750611, 8.869264060122936834528653001742, 10.00236502772710495631522690741, 11.40483854779093734319252940183, 11.61140015041932613004452031230

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