↑

Properties

Label 2-224-224.83-c1-0-14
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

Related objects

Downloads

Learn more

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: $\chi_{224} (83, \cdot )$
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 \( 1 - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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