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Properties

Label 2-224-224.27-c1-0-6
Degree $2$
Conductor $224$
Sign $0.734 - 0.678i$
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  + (−1.28 + 0.581i)2-s + (1.32 − 1.50i)4-s + (−1.87 + 1.87i)7-s + (−0.832 + 2.70i)8-s + (2.12 − 2.12i)9-s + (5.93 + 2.45i)11-s + (1.32 − 3.5i)14-s + (−0.5 − 3.96i)16-s + (−1.5 + 3.96i)18-s + (−9.07 + 0.284i)22-s + (6.64 + 6.64i)23-s + (3.53 + 3.53i)25-s + (0.331 + 5.28i)28-s + (−0.842 − 2.03i)29-s + (2.95 + 4.82i)32-s + ⋯
L(s)  = 1  + (−0.911 + 0.411i)2-s + (0.661 − 0.750i)4-s + (−0.707 + 0.707i)7-s + (−0.294 + 0.955i)8-s + (0.707 − 0.707i)9-s + (1.78 + 0.740i)11-s + (0.353 − 0.935i)14-s + (−0.125 − 0.992i)16-s + (−0.353 + 0.935i)18-s + (−1.93 + 0.0606i)22-s + (1.38 + 1.38i)23-s + (0.707 + 0.707i)25-s + (0.0626 + 0.998i)28-s + (−0.156 − 0.377i)29-s + (0.522 + 0.852i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804800 + 0.314797i\)
\(L(\frac12)\) \(\approx\) \(0.804800 + 0.314797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.581i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-5.93 - 2.45i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.64 - 6.64i)T + 23iT^{2} \)
29 \( 1 + (0.842 + 2.03i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (6.91 + 2.86i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + 41iT^{2} \)
43 \( 1 + (12.1 + 5.01i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-5.20 + 12.5i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-9.02 + 3.73i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (11.3 - 11.3i)T - 71iT^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + (58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.10685165673542228405866247579, −11.46409697364858045212921178121, −10.00056783063265300259537200768, −9.353669908894632519723487591809, −8.770591877156174885360492814313, −7.00825962209020362606327171514, −6.74540454295263609856360287765, −5.34834782844325160647868507261, −3.54325725377358678953532628008, −1.54202712703220817384561886071, 1.20034594838714165189656313292, 3.14603512836084835542216280942, 4.36192377352054413270021323128, 6.54087118713498603593349563891, 7.02038332861250472374588991199, 8.433712016807024547140203148429, 9.222545717171625468945957385812, 10.28443075155224839953592983139, 10.90720048113518715357521723445, 12.00749849973260602007955687683

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