↑

Properties

Label 2-224-224.195-c1-0-25
Degree $2$
Conductor $224$
Sign $-0.938 + 0.345i$
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  + (−0.581 − 1.28i)2-s + (−1.32 + 1.50i)4-s + (−1.87 − 1.87i)7-s + (2.70 + 0.832i)8-s + (−2.12 − 2.12i)9-s + (−2.18 − 5.28i)11-s + (−1.32 + 3.5i)14-s + (−0.5 − 3.96i)16-s + (−1.5 + 3.96i)18-s + (−5.53 + 5.89i)22-s + (1.35 − 1.35i)23-s + (−3.53 + 3.53i)25-s + (5.28 − 0.331i)28-s + (8.32 + 3.44i)29-s + (−4.82 + 2.95i)32-s + ⋯
L(s)  = 1  + (−0.411 − 0.911i)2-s + (−0.661 + 0.750i)4-s + (−0.707 − 0.707i)7-s + (0.955 + 0.294i)8-s + (−0.707 − 0.707i)9-s + (−0.660 − 1.59i)11-s + (−0.353 + 0.935i)14-s + (−0.125 − 0.992i)16-s + (−0.353 + 0.935i)18-s + (−1.18 + 1.25i)22-s + (0.282 − 0.282i)23-s + (−0.707 + 0.707i)25-s + (0.998 − 0.0626i)28-s + (1.54 + 0.640i)29-s + (−0.852 + 0.522i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107349 - 0.601866i\)
\(L(\frac12)\) \(\approx\) \(0.107349 - 0.601866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.581 + 1.28i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (2.18 + 5.28i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.35 + 1.35i)T - 23iT^{2} \)
29 \( 1 + (-8.32 - 3.44i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (0.571 + 1.37i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 - 41iT^{2} \)
43 \( 1 + (3.62 + 8.75i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-12.2 + 5.08i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-6.19 + 14.9i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-11.3 - 11.3i)T + 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
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

−11.66538237814201986447303302072, −10.84639422797237364361901264391, −10.04900082784849210217663660752, −8.933238443983748010344152037089, −8.216251817485463207542430108889, −6.83181330436382249130276462720, −5.47625877347543116652107837257, −3.74269103107780975951701324299, −2.93591725323949068662120431262, −0.57107601502338766970294908635, 2.43085086568258066445972264357, 4.58447805543994516638845410126, 5.58114310439395255670317116264, 6.64851487826420318302340476206, 7.74695686011973175480704579470, 8.602985387813530746173486562654, 9.747161080039507603058119572296, 10.31630119826219272592524889379, 11.77718185498497103906052355512, 12.84294455811222496391388686486

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