↑

Properties

Label 2-74-37.27-c3-0-5
Degree $2$
Conductor $74$
Sign $0.413 - 0.910i$
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 + (3.54 + 6.13i)3-s + (1.99 + 3.46i)4-s + (13.1 − 7.57i)5-s + 14.1i·6-s + (−5.28 − 9.16i)7-s + 7.99i·8-s + (−11.6 + 20.1i)9-s + 30.3·10-s − 26.3·11-s + (−14.1 + 24.5i)12-s + (−38.4 + 22.1i)13-s − 21.1i·14-s + (93.0 + 53.7i)15-s + (−8 + 13.8i)16-s + (−65.3 − 37.7i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.682 + 1.18i)3-s + (0.249 + 0.433i)4-s + (1.17 − 0.677i)5-s + 0.964i·6-s + (−0.285 − 0.494i)7-s + 0.353i·8-s + (−0.430 + 0.745i)9-s + 0.958·10-s − 0.723·11-s + (−0.341 + 0.590i)12-s + (−0.820 + 0.473i)13-s − 0.403i·14-s + (1.60 + 0.924i)15-s + (−0.125 + 0.216i)16-s + (−0.932 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.413 - 0.910i$
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: \((2,\ 74,\ (\ :3/2),\ 0.413 - 0.910i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.22940 + 1.43570i\)
\(L(\frac12)\) \(\approx\) \(2.22940 + 1.43570i\)
\(L(\frac{5}{2})\) 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 \( 1 + (85.8 + 208. i)T \)
good3 \( 1 + (-3.54 - 6.13i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-13.1 + 7.57i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (5.28 + 9.16i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 26.3T + 1.33e3T^{2} \)
13 \( 1 + (38.4 - 22.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (65.3 + 37.7i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-90.0 + 51.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 43.3iT - 1.21e4T^{2} \)
29 \( 1 + 78.7iT - 2.43e4T^{2} \)
31 \( 1 - 264. iT - 2.97e4T^{2} \)
41 \( 1 + (89.5 + 155. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 208. iT - 7.95e4T^{2} \)
47 \( 1 - 529.T + 1.03e5T^{2} \)
53 \( 1 + (307. - 532. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-340. - 196. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (650. - 375. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (424. + 735. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-508. - 881. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 653.T + 3.89e5T^{2} \)
79 \( 1 + (716. - 413. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-67.5 + 116. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (302. + 174. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 95.4iT - 9.12e5T^{2} \)
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

−14.04446598700932418350459982554, −13.66195262211231973844966007330, −12.41616476035014644772871059765, −10.66107687604021377547177091837, −9.611824545812889573985530241646, −8.874663482691728179562017480099, −7.10407135407006795023167945764, −5.35697310695377500184977327810, −4.43809755580515505533547782641, −2.70267555886191952570679843889, 1.98850669221901299282081013191, 2.89380939232994738827470433524, 5.49850006643701288877651409479, 6.61562881847070261187479430935, 7.83079959215847635800699393674, 9.475940322339885813786296823907, 10.52931366010696344667075880155, 12.08039147939662750605497448908, 13.08624643228517054020340246719, 13.62889528145534358798217099741

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