Properties

Label 2-224-32.13-c1-0-23
Degree $2$
Conductor $224$
Sign $-0.802 + 0.596i$
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  + (0.629 − 1.26i)2-s + (0.722 − 1.74i)3-s + (−1.20 − 1.59i)4-s + (−1.56 + 0.647i)5-s + (−1.75 − 2.01i)6-s + (0.707 − 0.707i)7-s + (−2.77 + 0.526i)8-s + (−0.401 − 0.401i)9-s + (−0.163 + 2.38i)10-s + (−1.93 − 4.67i)11-s + (−3.65 + 0.955i)12-s + (4.00 + 1.65i)13-s + (−0.450 − 1.34i)14-s + 3.19i·15-s + (−1.08 + 3.85i)16-s − 0.601i·17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (0.417 − 1.00i)3-s + (−0.603 − 0.797i)4-s + (−0.698 + 0.289i)5-s + (−0.716 − 0.822i)6-s + (0.267 − 0.267i)7-s + (−0.982 + 0.186i)8-s + (−0.133 − 0.133i)9-s + (−0.0517 + 0.754i)10-s + (−0.584 − 1.41i)11-s + (−1.05 + 0.275i)12-s + (1.11 + 0.460i)13-s + (−0.120 − 0.358i)14-s + 0.824i·15-s + (−0.270 + 0.962i)16-s − 0.145i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.802 + 0.596i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459306 - 1.38758i\)
\(L(\frac12)\) \(\approx\) \(0.459306 - 1.38758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.629 + 1.26i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.722 + 1.74i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.56 - 0.647i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (1.93 + 4.67i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.00 - 1.65i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.601iT - 17T^{2} \)
19 \( 1 + (-1.47 - 0.612i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.84 - 2.84i)T + 23iT^{2} \)
29 \( 1 + (-0.757 + 1.82i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.06T + 31T^{2} \)
37 \( 1 + (1.69 - 0.700i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.24 + 2.24i)T + 41iT^{2} \)
43 \( 1 + (3.46 + 8.35i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.14iT - 47T^{2} \)
53 \( 1 + (2.13 + 5.15i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-8.00 + 3.31i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.73 - 9.02i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.04 - 12.1i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (8.56 - 8.56i)T - 71iT^{2} \)
73 \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \)
79 \( 1 - 1.33iT - 79T^{2} \)
83 \( 1 + (0.0908 + 0.0376i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.11 - 3.11i)T - 89iT^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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

−11.71478359002903791967649447052, −11.26340556117358165599307449869, −10.26050685806070959654632450561, −8.726590043503517724000082317016, −8.041828763131454946603641434024, −6.79583025729163773522972823301, −5.53345612150327880541891919173, −3.93825601102302288961938310043, −2.86709167909356535098680999743, −1.18857991508325274173903429619, 3.19574074105838647449966550787, 4.37236470238430282663286869667, 4.99963975504838507872506254138, 6.53417791443030366802907739456, 7.81370359490072006963504627977, 8.525929276652611525344413680439, 9.514936742113650429442729360944, 10.54686143613814454637372342041, 11.93387365415323527530452598916, 12.71081892372126793789689051159

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