Properties

Label 2-74-37.6-c6-0-18
Degree $2$
Conductor $74$
Sign $-0.388 - 0.921i$
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 − 12.3i·3-s − 32i·4-s + (−142. − 142. i)5-s + (−49.4 − 49.4i)6-s − 327.·7-s + (−128 − 128i)8-s + 576.·9-s − 1.13e3·10-s + 1.87e3i·11-s − 395.·12-s + (471. + 471. i)13-s + (−1.30e3 + 1.30e3i)14-s + (−1.75e3 + 1.75e3i)15-s − 1.02e3·16-s + (1.65e3 + 1.65e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.457i·3-s − 0.5i·4-s + (−1.13 − 1.13i)5-s + (−0.228 − 0.228i)6-s − 0.954·7-s + (−0.250 − 0.250i)8-s + 0.790·9-s − 1.13·10-s + 1.40i·11-s − 0.228·12-s + (0.214 + 0.214i)13-s + (−0.477 + 0.477i)14-s + (−0.520 + 0.520i)15-s − 0.250·16-s + (0.336 + 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.388 - 0.921i$
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: \((2,\ 74,\ (\ :3),\ -0.388 - 0.921i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.194187 + 0.292512i\)
\(L(\frac12)\) \(\approx\) \(0.194187 + 0.292512i\)
\(L(4)\) 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 \( 1 + (2.34e4 + 4.48e4i)T \)
good3 \( 1 + 12.3iT - 729T^{2} \)
5 \( 1 + (142. + 142. i)T + 1.56e4iT^{2} \)
7 \( 1 + 327.T + 1.17e5T^{2} \)
11 \( 1 - 1.87e3iT - 1.77e6T^{2} \)
13 \( 1 + (-471. - 471. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-1.65e3 - 1.65e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (331. + 331. i)T + 4.70e7iT^{2} \)
23 \( 1 + (1.65e4 + 1.65e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (4.64e3 - 4.64e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (2.83e4 - 2.83e4i)T - 8.87e8iT^{2} \)
41 \( 1 - 7.09e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.85e4 + 2.85e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 4.76e4T + 1.07e10T^{2} \)
53 \( 1 - 1.67e4T + 2.21e10T^{2} \)
59 \( 1 + (1.60e5 + 1.60e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.23e5 - 1.23e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.32e4iT - 9.04e10T^{2} \)
71 \( 1 - 3.32e5T + 1.28e11T^{2} \)
73 \( 1 + 6.56e5iT - 1.51e11T^{2} \)
79 \( 1 + (5.70e5 + 5.70e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 7.17e5T + 3.26e11T^{2} \)
89 \( 1 + (1.76e5 - 1.76e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-1.15e6 - 1.15e6i)T + 8.32e11iT^{2} \)
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

−12.40479241315099643467011567356, −12.18286232985742332159527105668, −10.40432596132716976002414157840, −9.273037313975595394563628195039, −7.82579442487273718606101826305, −6.60842470466384977472519416305, −4.74414250899483247076335987750, −3.80855792088860023222785819819, −1.69403734275696581328741553371, −0.11294397825552101396264758071, 3.27985972750267149448050929051, 3.85153118322949861252021096800, 5.84168630413271452356125719376, 7.02509284207138451303547739541, 8.031298080141289892418670709683, 9.698768272042949027898626558435, 10.91299561159258060400561826169, 11.86205687604654831278256254500, 13.20779117696895508686421454058, 14.20691346883348368208751226927

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