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Properties

Label 2-224-32.5-c1-0-23
Degree $2$
Conductor $224$
Sign $-0.831 - 0.555i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1 − 2.41i)3-s − 2i·4-s + (−3.41 − 1.41i)5-s + (3.41 + 1.41i)6-s + (0.707 + 0.707i)7-s + (2 + 2i)8-s + (−2.70 + 2.70i)9-s + (4.82 − 1.99i)10-s + (−2.29 + 5.53i)11-s + (−4.82 + 2i)12-s + (1.41 − 0.585i)13-s − 1.41·14-s + 9.65i·15-s − 4·16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.577 − 1.39i)3-s − i·4-s + (−1.52 − 0.632i)5-s + (1.39 + 0.577i)6-s + (0.267 + 0.267i)7-s + (0.707 + 0.707i)8-s + (−0.902 + 0.902i)9-s + (1.52 − 0.632i)10-s + (−0.691 + 1.66i)11-s + (−1.39 + 0.577i)12-s + (0.392 − 0.162i)13-s − 0.377·14-s + 2.49i·15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1 + 2.41i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.41 + 1.41i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (2.29 - 5.53i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.41 + 0.585i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (-0.414 + 0.171i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (3.29 + 7.94i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + (4.70 + 1.94i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.41 - 3.41i)T - 41iT^{2} \)
43 \( 1 + (1.87 - 4.53i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.82iT - 47T^{2} \)
53 \( 1 + (-0.0502 + 0.121i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5 + 2.07i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.17 + 2.82i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.70 - 11.3i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (4.24 + 4.24i)T + 71iT^{2} \)
73 \( 1 + (-3.17 + 3.17i)T - 73iT^{2} \)
79 \( 1 + 9.89iT - 79T^{2} \)
83 \( 1 + (-3.24 + 1.34i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.17 + 5.17i)T + 89iT^{2} \)
97 \( 1 - 1.17T + 97T^{2} \)
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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

−11.76541849052425798311231948687, −10.94552592262718789301787744651, −9.476213480338880563508409335678, −8.153882298323288257339066305161, −7.66736408639700318581125906583, −7.00839530588096599093699724654, −5.65424998802923106598365758339, −4.52918745644796550103106267128, −1.73567312311397554560939721570, 0, 3.40735467730031703534106433827, 3.79388063117807347582304027346, 5.25860399477540022808461512775, 7.06858680240611138633294870478, 8.203221857541121846376713502020, 8.957861858034234570810244812229, 10.40892057551483987428811380039, 10.93476585625036152819591588905, 11.27414583215121316231993154319

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