Properties

Label 2-224-32.29-c1-0-11
Degree $2$
Conductor $224$
Sign $0.847 + 0.530i$
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  + (−0.533 + 1.30i)2-s + (−0.910 − 0.377i)3-s + (−1.43 − 1.39i)4-s + (−0.227 − 0.549i)5-s + (0.979 − 0.991i)6-s + (−0.707 + 0.707i)7-s + (2.59 − 1.12i)8-s + (−1.43 − 1.43i)9-s + (0.840 − 0.00493i)10-s + (4.11 − 1.70i)11-s + (0.775 + 1.81i)12-s + (2.17 − 5.25i)13-s + (−0.548 − 1.30i)14-s + 0.586i·15-s + (0.0938 + 3.99i)16-s − 4.07i·17-s + ⋯
L(s)  = 1  + (−0.377 + 0.926i)2-s + (−0.525 − 0.217i)3-s + (−0.715 − 0.698i)4-s + (−0.101 − 0.245i)5-s + (0.400 − 0.404i)6-s + (−0.267 + 0.267i)7-s + (0.917 − 0.398i)8-s + (−0.478 − 0.478i)9-s + (0.265 − 0.00155i)10-s + (1.23 − 0.513i)11-s + (0.223 + 0.523i)12-s + (0.604 − 1.45i)13-s + (−0.146 − 0.348i)14-s + 0.151i·15-s + (0.0234 + 0.999i)16-s − 0.988i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.847 + 0.530i$
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: \((2,\ 224,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668742 - 0.192200i\)
\(L(\frac12)\) \(\approx\) \(0.668742 - 0.192200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.533 - 1.30i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.910 + 0.377i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.227 + 0.549i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (-4.11 + 1.70i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.17 + 5.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 4.07iT - 17T^{2} \)
19 \( 1 + (0.900 - 2.17i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.759 + 0.759i)T + 23iT^{2} \)
29 \( 1 + (2.59 + 1.07i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 5.93T + 31T^{2} \)
37 \( 1 + (-4.05 - 9.78i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.57 + 7.57i)T + 41iT^{2} \)
43 \( 1 + (-2.08 + 0.864i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.52iT - 47T^{2} \)
53 \( 1 + (-4.99 + 2.06i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.25 + 3.02i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.79 + 1.98i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-7.30 - 3.02i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (7.81 - 7.81i)T - 71iT^{2} \)
73 \( 1 + (-8.65 - 8.65i)T + 73iT^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 + (-2.50 + 6.05i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.86 + 6.86i)T - 89iT^{2} \)
97 \( 1 - 0.269T + 97T^{2} \)
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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

−12.18134311649484038554367421995, −11.20493533406501362355534480765, −10.05731546651358146223759683356, −8.953783120877709576534170625665, −8.314149959845929106907499957524, −6.92687436360356391715442625473, −6.07494070775526024937524121597, −5.27880900329925395974145137588, −3.59681257081688624689665504257, −0.75199280668690443628240543398, 1.79225754082998243082099733595, 3.64641923832735779655998871580, 4.58004817843794707191034695272, 6.22401232003830314555500003904, 7.38656088937871283353779072027, 8.824730323063624831287252964291, 9.435065228052067457946218885739, 10.73943543878058751355227349330, 11.20839794472346467798551264490, 12.03512339357987881927516651578

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