↑

Properties

Label 2-74-37.6-c6-0-5
Degree $2$
Conductor $74$
Sign $0.999 + 0.00713i$
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 − 47.9i·3-s − 32i·4-s + (99.4 + 99.4i)5-s + (191. + 191. i)6-s + 584.·7-s + (128 + 128i)8-s − 1.56e3·9-s − 795.·10-s + 2.32e3i·11-s − 1.53e3·12-s + (510. + 510. i)13-s + (−2.33e3 + 2.33e3i)14-s + (4.76e3 − 4.76e3i)15-s − 1.02e3·16-s + (2.28e3 + 2.28e3i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 1.77i·3-s − 0.5i·4-s + (0.795 + 0.795i)5-s + (0.887 + 0.887i)6-s + 1.70·7-s + (0.250 + 0.250i)8-s − 2.14·9-s − 0.795·10-s + 1.74i·11-s − 0.887·12-s + (0.232 + 0.232i)13-s + (−0.851 + 0.851i)14-s + (1.41 − 1.41i)15-s − 0.250·16-s + (0.465 + 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00713i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.00713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.999 + 0.00713i$
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: \((2,\ 74,\ (\ :3),\ 0.999 + 0.00713i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.93870 - 0.00691393i\)
\(L(\frac12)\) \(\approx\) \(1.93870 - 0.00691393i\)
\(L(4)\) 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 \( 1 + (5.05e4 - 3.81e3i)T \)
good3 \( 1 + 47.9iT - 729T^{2} \)
5 \( 1 + (-99.4 - 99.4i)T + 1.56e4iT^{2} \)
7 \( 1 - 584.T + 1.17e5T^{2} \)
11 \( 1 - 2.32e3iT - 1.77e6T^{2} \)
13 \( 1 + (-510. - 510. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-2.28e3 - 2.28e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-4.48e3 - 4.48e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-6.69e3 - 6.69e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (9.38e3 - 9.38e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (-3.74e4 + 3.74e4i)T - 8.87e8iT^{2} \)
41 \( 1 + 2.03e4iT - 4.75e9T^{2} \)
43 \( 1 + (3.22e4 + 3.22e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.80e5T + 1.07e10T^{2} \)
53 \( 1 - 8.48e4T + 2.21e10T^{2} \)
59 \( 1 + (1.57e5 + 1.57e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-2.27e5 + 2.27e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 1.25e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.76e5T + 1.28e11T^{2} \)
73 \( 1 + 1.63e5iT - 1.51e11T^{2} \)
79 \( 1 + (1.09e5 + 1.09e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 4.31e5T + 3.26e11T^{2} \)
89 \( 1 + (5.79e5 - 5.79e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-8.65e5 - 8.65e5i)T + 8.32e11iT^{2} \)
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

−13.59959988746461779721768244241, −12.25069615725399990886813969779, −11.27211478022837402193529166751, −9.895778398761320217826584317319, −8.221998071724291790667908141243, −7.46977766048945937024412043402, −6.58893905115491302094888077384, −5.28396408397577634885095000918, −2.05530766262115473821181112918, −1.46661584109450469508137970793, 1.03367929832561135068000521071, 3.12363609657539183566207306606, 4.75500592857174526363728145907, 5.46548348559891159780235308708, 8.384645933219110236663366860993, 8.835630988688534470247632503964, 10.01843750030626854368021594033, 11.02278949975402431001721191348, 11.61634342600332007215136961406, 13.60285677458004365639058570937

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