↑

Properties

Label 2-74-37.10-c7-0-17
Degree $2$
Conductor $74$
Sign $-0.864 + 0.502i$
Analytic cond. $23.1164$
Root an. cond. $4.80796$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (34.1 + 59.2i)3-s + (−31.9 − 55.4i)4-s + (13.0 + 22.6i)5-s + 546.·6-s + (−837. − 1.45e3i)7-s − 511.·8-s + (−1.24e3 + 2.15e3i)9-s + 208.·10-s − 5.50e3·11-s + (2.18e3 − 3.78e3i)12-s + (3.58e3 + 6.21e3i)13-s − 1.34e4·14-s + (−892. + 1.54e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (509. − 883. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.730 + 1.26i)3-s + (−0.249 − 0.433i)4-s + (0.0466 + 0.0808i)5-s + 1.03·6-s + (−0.923 − 1.59i)7-s − 0.353·8-s + (−0.568 + 0.985i)9-s + 0.0660·10-s − 1.24·11-s + (0.365 − 0.633i)12-s + (0.453 + 0.784i)13-s − 1.30·14-s + (−0.0682 + 0.118i)15-s + (−0.125 + 0.216i)16-s + (0.0251 − 0.0435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.864 + 0.502i$
Analytic conductor: \(23.1164\)
Root analytic conductor: \(4.80796\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :7/2),\ -0.864 + 0.502i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.229878 - 0.852599i\)
\(L(\frac12)\) \(\approx\) \(0.229878 - 0.852599i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 6.92i)T \)
37 \( 1 + (-2.86e5 + 1.14e5i)T \)
good3 \( 1 + (-34.1 - 59.2i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-13.0 - 22.6i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (837. + 1.45e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 5.50e3T + 1.94e7T^{2} \)
13 \( 1 + (-3.58e3 - 6.21e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-509. + 883. i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.69e4 + 4.67e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 1.06e5T + 3.40e9T^{2} \)
29 \( 1 + 6.96e4T + 1.72e10T^{2} \)
31 \( 1 - 1.61e5T + 2.75e10T^{2} \)
41 \( 1 + (-3.50e4 - 6.06e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 5.96e5T + 2.71e11T^{2} \)
47 \( 1 + 1.09e6T + 5.06e11T^{2} \)
53 \( 1 + (-1.79e5 + 3.10e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.38e6 + 2.39e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.38e6 - 2.40e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.30e6 - 2.25e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (1.63e5 + 2.83e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 2.90e6T + 1.10e13T^{2} \)
79 \( 1 + (-3.32e5 - 5.76e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.29e6 + 2.24e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (2.19e6 - 3.79e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 1.04e6T + 8.07e13T^{2} \)
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

−13.05885652368321602763072045436, −11.17940212893923547643854832067, −10.23547655600936257405001763470, −9.774447699959732133396469876034, −8.339638234860288968544222036435, −6.61681056783653010291888318701, −4.60105894034061668956106717707, −3.85322733940656867632648441464, −2.62905330795704709210735565687, −0.22376852600119187155226881562, 2.10316931988818556844018319946, 3.20062714827599164440517657547, 5.61408462412428829432984451682, 6.38410084061066073926577466461, 8.002045301139197254474714184916, 8.401782421455513318649939752053, 9.944431478748567948985240383450, 12.05142190235471962304991223187, 12.83958039609908379127755367542, 13.29515245160998271824835178101

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