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Properties

Label 2-224-7.2-c1-0-5
Degree $2$
Conductor $224$
Sign $0.827 + 0.561i$
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  + (0.207 + 0.358i)3-s + (0.914 − 1.58i)5-s + (−1 − 2.44i)7-s + (1.41 − 2.44i)9-s + (1.20 + 2.09i)11-s + 2.82·13-s + 0.757·15-s + (0.0857 + 0.148i)17-s + (−3.20 + 5.55i)19-s + (0.671 − 0.866i)21-s + (2.62 − 4.54i)23-s + (0.828 + 1.43i)25-s + 2.41·27-s − 2.82·29-s + (−2.79 − 4.83i)31-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s + (0.408 − 0.708i)5-s + (−0.377 − 0.925i)7-s + (0.471 − 0.816i)9-s + (0.363 + 0.630i)11-s + 0.784·13-s + 0.195·15-s + (0.0208 + 0.0360i)17-s + (−0.735 + 1.27i)19-s + (0.146 − 0.188i)21-s + (0.546 − 0.946i)23-s + (0.165 + 0.286i)25-s + 0.464·27-s − 0.525·29-s + (−0.501 − 0.868i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28976 - 0.396304i\)
\(L(\frac12)\) \(\approx\) \(1.28976 - 0.396304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good3 \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.914 + 1.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.20 - 2.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (-0.0857 - 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.20 - 5.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.62 + 4.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + (2.79 + 4.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.32 - 7.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 + (5.20 - 9.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.32 - 7.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.37 - 2.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (-7.32 - 12.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
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

−12.53597386128235055752458637300, −11.07843385034119615610633324111, −10.05561745447979236665110637190, −9.377272580032704368484904457018, −8.342343159198228379869204235707, −7.00131670928211950491475012751, −6.08565915592146944576062921415, −4.52895133251855484761339751576, −3.62707871477226001089143521821, −1.36727147017300784984704245039, 2.10171973999831898629061990399, 3.39169220225769022300557523908, 5.17037375587120897371023224091, 6.28673906963438353955453097504, 7.16440719309112648104559004316, 8.553638796794282041302869268102, 9.290647499409195154534740801984, 10.61862773109352978610824818578, 11.18127161619520297945610427687, 12.45651519029387809705632477844

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