↑

Properties

Label 2-74-37.27-c3-0-8
Degree $2$
Conductor $74$
Sign $-0.973 + 0.227i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−3.38 − 5.86i)3-s + (1.99 + 3.46i)4-s + (14.5 − 8.42i)5-s + 13.5i·6-s + (−7.56 − 13.0i)7-s − 7.99i·8-s + (−9.43 + 16.3i)9-s − 33.7·10-s − 16.4·11-s + (13.5 − 23.4i)12-s + (−75.3 + 43.5i)13-s + 30.2i·14-s + (−98.8 − 57.0i)15-s + (−8 + 13.8i)16-s + (−1.45 − 0.841i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.651 − 1.12i)3-s + (0.249 + 0.433i)4-s + (1.30 − 0.753i)5-s + 0.921i·6-s + (−0.408 − 0.707i)7-s − 0.353i·8-s + (−0.349 + 0.605i)9-s − 1.06·10-s − 0.450·11-s + (0.325 − 0.564i)12-s + (−1.60 + 0.928i)13-s + 0.577i·14-s + (−1.70 − 0.982i)15-s + (−0.125 + 0.216i)16-s + (−0.0208 − 0.0120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.973 + 0.227i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.973 + 0.227i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0949029 - 0.823560i\)
\(L(\frac12)\) \(\approx\) \(0.0949029 - 0.823560i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
37 \( 1 + (-112. + 194. i)T \)
good3 \( 1 + (3.38 + 5.86i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-14.5 + 8.42i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (7.56 + 13.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 16.4T + 1.33e3T^{2} \)
13 \( 1 + (75.3 - 43.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.45 + 0.841i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-108. + 62.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 85.4iT - 1.21e4T^{2} \)
29 \( 1 - 151. iT - 2.43e4T^{2} \)
31 \( 1 + 88.7iT - 2.97e4T^{2} \)
41 \( 1 + (-217. - 375. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 215. iT - 7.95e4T^{2} \)
47 \( 1 + 101.T + 1.03e5T^{2} \)
53 \( 1 + (-249. + 431. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (577. + 333. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-660. + 381. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-87.8 - 152. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-214. - 370. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 162.T + 3.89e5T^{2} \)
79 \( 1 + (-476. + 275. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-407. + 705. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (290. + 167. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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

−13.17082981790561060117244012816, −12.59490569502821709420660031523, −11.48248248984228896622739414274, −10.00113890093097578661408330839, −9.271825648075850377916942107487, −7.49097877935753700599131046248, −6.59995850032886596671535608554, −5.09942525718124905311255375744, −2.17564356968941267615805934355, −0.68534453421445079034186178876, 2.68304178876794630735221470457, 5.28763497811987891023286179154, 5.89531039580585835891035945403, 7.53933510553384476047177980887, 9.587652118214743579390115888006, 9.842693784208569195195633882388, 10.75434982375519537242910378030, 12.16533675507739727881773395911, 13.76008673225048943335004470887, 14.95786103465812431771732762418

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