Properties

Label 2-224-32.29-c1-0-14
Degree $2$
Conductor $224$
Sign $0.938 + 0.344i$
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  + (1.16 − 0.795i)2-s + (1.29 + 0.536i)3-s + (0.735 − 1.85i)4-s + (0.937 + 2.26i)5-s + (1.94 − 0.402i)6-s + (−0.707 + 0.707i)7-s + (−0.619 − 2.75i)8-s + (−0.730 − 0.730i)9-s + (2.89 + 1.90i)10-s + (−2.12 + 0.879i)11-s + (1.95 − 2.01i)12-s + (−0.267 + 0.644i)13-s + (−0.264 + 1.38i)14-s + 3.43i·15-s + (−2.91 − 2.73i)16-s − 2.83i·17-s + ⋯
L(s)  = 1  + (0.826 − 0.562i)2-s + (0.748 + 0.309i)3-s + (0.367 − 0.929i)4-s + (0.419 + 1.01i)5-s + (0.792 − 0.164i)6-s + (−0.267 + 0.267i)7-s + (−0.218 − 0.975i)8-s + (−0.243 − 0.243i)9-s + (0.916 + 0.601i)10-s + (−0.640 + 0.265i)11-s + (0.563 − 0.581i)12-s + (−0.0740 + 0.178i)13-s + (−0.0707 + 0.371i)14-s + 0.887i·15-s + (−0.729 − 0.683i)16-s − 0.687i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.938 + 0.344i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20584 - 0.392307i\)
\(L(\frac12)\) \(\approx\) \(2.20584 - 0.392307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.795i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-1.29 - 0.536i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.937 - 2.26i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (2.12 - 0.879i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.267 - 0.644i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.83iT - 17T^{2} \)
19 \( 1 + (0.445 - 1.07i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.15 + 4.15i)T + 23iT^{2} \)
29 \( 1 + (-3.37 - 1.39i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 2.27T + 31T^{2} \)
37 \( 1 + (-3.59 - 8.67i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.419 - 0.419i)T + 41iT^{2} \)
43 \( 1 + (6.63 - 2.74i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 + (-8.31 + 3.44i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.36 + 8.12i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-8.49 - 3.52i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.08 - 1.69i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.89 - 9.89i)T - 71iT^{2} \)
73 \( 1 + (-1.37 - 1.37i)T + 73iT^{2} \)
79 \( 1 + 3.67iT - 79T^{2} \)
83 \( 1 + (-2.34 + 5.64i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (11.4 - 11.4i)T - 89iT^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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

−12.21615566076415315831159456582, −11.30621780436473353779613677342, −10.14410908465098124711890165906, −9.751468083392642528808479409507, −8.360707101118424746950357741013, −6.82915071499521567092636375108, −5.98258591785543977133084322564, −4.53882487445862753624073288441, −3.10629274548672198147374213982, −2.47215310939391468245211558139, 2.25417846368083165681867285238, 3.68298164016309748250272307555, 5.07648885848302870701364080603, 5.94090540311715409832129600422, 7.38003262430693106905142500713, 8.249628390533251956968130911128, 8.951773670018705533335317744546, 10.36932138603308859817712675631, 11.70867105293060212863811618571, 12.76943055913582738515416142734

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