Properties

Label 2-74-37.6-c6-0-17
Degree $2$
Conductor $74$
Sign $-0.741 - 0.671i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
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 − 4i)2-s − 42.5i·3-s − 32i·4-s + (−72.1 − 72.1i)5-s + (−170. − 170. i)6-s + 66.7·7-s + (−128 − 128i)8-s − 1.07e3·9-s − 577.·10-s − 1.34e3i·11-s − 1.36e3·12-s + (906. + 906. i)13-s + (266. − 266. i)14-s + (−3.07e3 + 3.07e3i)15-s − 1.02e3·16-s + (939. + 939. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 1.57i·3-s − 0.5i·4-s + (−0.577 − 0.577i)5-s + (−0.787 − 0.787i)6-s + 0.194·7-s + (−0.250 − 0.250i)8-s − 1.48·9-s − 0.577·10-s − 1.01i·11-s − 0.787·12-s + (0.412 + 0.412i)13-s + (0.0972 − 0.0972i)14-s + (−0.909 + 0.909i)15-s − 0.250·16-s + (0.191 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ -0.741 - 0.671i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.615257 + 1.59674i\)
\(L(\frac12)\) \(\approx\) \(0.615257 + 1.59674i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 4i)T \)
37 \( 1 + (4.02e4 + 3.07e4i)T \)
good3 \( 1 + 42.5iT - 729T^{2} \)
5 \( 1 + (72.1 + 72.1i)T + 1.56e4iT^{2} \)
7 \( 1 - 66.7T + 1.17e5T^{2} \)
11 \( 1 + 1.34e3iT - 1.77e6T^{2} \)
13 \( 1 + (-906. - 906. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-939. - 939. i)T + 2.41e7iT^{2} \)
19 \( 1 + (203. + 203. i)T + 4.70e7iT^{2} \)
23 \( 1 + (-1.42e4 - 1.42e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (1.70e4 - 1.70e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (-4.66e3 + 4.66e3i)T - 8.87e8iT^{2} \)
41 \( 1 + 5.62e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.06e4 + 2.06e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 6.20e4T + 1.07e10T^{2} \)
53 \( 1 + 1.74e4T + 2.21e10T^{2} \)
59 \( 1 + (-6.99e4 - 6.99e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (-2.96e5 + 2.96e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 4.23e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.50e4T + 1.28e11T^{2} \)
73 \( 1 + 1.43e5iT - 1.51e11T^{2} \)
79 \( 1 + (-1.71e5 - 1.71e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 2.76e5T + 3.26e11T^{2} \)
89 \( 1 + (-5.07e5 + 5.07e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (6.43e5 + 6.43e5i)T + 8.32e11iT^{2} \)
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

−12.72992268341383200794533486278, −11.79785353945762598315177778905, −11.02996868413764201912689279361, −8.964173514875331388881336970693, −7.920754239823482631635421431971, −6.66184392158689994682723200046, −5.34699507347094137437642431433, −3.47636869207082271968241135457, −1.70654768934153231404873905307, −0.56910953192644054211298579341, 3.08723392814227677509249000118, 4.23647082617308288754478769250, 5.21817950382734903102053149550, 6.89496439641038041367091454150, 8.292718120654076382385480057246, 9.628797653493908502511088812532, 10.70141688243385646609990012559, 11.63625400476289219837352390856, 13.07084275251316121641530634668, 14.71393719865212450019567524787

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