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Properties

Label 2-224-32.29-c1-0-15
Degree $2$
Conductor $224$
Sign $0.777 - 0.629i$
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  + (1.27 + 0.618i)2-s + (2.04 + 0.847i)3-s + (1.23 + 1.57i)4-s + (−1.31 − 3.17i)5-s + (2.07 + 2.34i)6-s + (−0.707 + 0.707i)7-s + (0.595 + 2.76i)8-s + (1.34 + 1.34i)9-s + (0.292 − 4.84i)10-s + (−3.26 + 1.35i)11-s + (1.19 + 4.26i)12-s + (1.07 − 2.58i)13-s + (−1.33 + 0.461i)14-s − 7.60i·15-s + (−0.953 + 3.88i)16-s − 3.99i·17-s + ⋯
L(s)  = 1  + (0.899 + 0.437i)2-s + (1.18 + 0.489i)3-s + (0.617 + 0.786i)4-s + (−0.587 − 1.41i)5-s + (0.848 + 0.957i)6-s + (−0.267 + 0.267i)7-s + (0.210 + 0.977i)8-s + (0.449 + 0.449i)9-s + (0.0923 − 1.53i)10-s + (−0.983 + 0.407i)11-s + (0.343 + 1.23i)12-s + (0.297 − 0.718i)13-s + (−0.357 + 0.123i)14-s − 1.96i·15-s + (−0.238 + 0.971i)16-s − 0.968i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.777 - 0.629i$
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: \((2,\ 224,\ (\ :1/2),\ 0.777 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25368 + 0.798301i\)
\(L(\frac12)\) \(\approx\) \(2.25368 + 0.798301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.618i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-2.04 - 0.847i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.31 + 3.17i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (3.26 - 1.35i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.07 + 2.58i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.99iT - 17T^{2} \)
19 \( 1 + (2.91 - 7.03i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.47 + 1.47i)T + 23iT^{2} \)
29 \( 1 + (2.10 + 0.872i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.19T + 31T^{2} \)
37 \( 1 + (-0.385 - 0.930i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.37 - 6.37i)T + 41iT^{2} \)
43 \( 1 + (-8.95 + 3.71i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.31iT - 47T^{2} \)
53 \( 1 + (-0.875 + 0.362i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.860 - 2.07i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (11.8 + 4.92i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.95 + 0.808i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.27 - 5.27i)T - 71iT^{2} \)
73 \( 1 + (-8.05 - 8.05i)T + 73iT^{2} \)
79 \( 1 - 9.04iT - 79T^{2} \)
83 \( 1 + (-4.16 + 10.0i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-12.4 + 12.4i)T - 89iT^{2} \)
97 \( 1 + 3.17T + 97T^{2} \)
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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

−12.61165628869451438330834062836, −11.87519300270343694655823630759, −10.31067829992269749383111061949, −9.120313087231501307333936999610, −8.148269510360791096779253063883, −7.80413376603517076232311775034, −5.89083937142540140704728047175, −4.76887988607684117258670672343, −3.86486415582767504556359113767, −2.60391504422537260090622323461, 2.37438619068920275691338740685, 3.11999474877593318291730382095, 4.21598140469592739325654072791, 6.14649575226308125333290190366, 7.08088948974743010607443517427, 7.903852554813230184008670506698, 9.266266924410230160177277571660, 10.77648725741321730674368431598, 10.93961719300264622393350979563, 12.34099672760855110513767066944

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