Properties

Label 2-74-37.33-c5-0-1
Degree $2$
Conductor $74$
Sign $0.440 - 0.897i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.694 − 3.93i)2-s + (−2.84 − 16.1i)3-s + (−15.0 − 5.47i)4-s + (−37.4 + 31.4i)5-s − 65.4·6-s + (−21.7 + 18.2i)7-s + (−32 + 55.4i)8-s + (−23.5 + 8.58i)9-s + (97.7 + 169. i)10-s + (−281. + 487. i)11-s + (−45.4 + 257. i)12-s + (−128. − 46.7i)13-s + (56.8 + 98.5i)14-s + (613. + 514. i)15-s + (196. + 164. i)16-s + (673. − 245. i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.182 − 1.03i)3-s + (−0.469 − 0.171i)4-s + (−0.669 + 0.562i)5-s − 0.742·6-s + (−0.168 + 0.141i)7-s + (−0.176 + 0.306i)8-s + (−0.0970 + 0.0353i)9-s + (0.309 + 0.535i)10-s + (−0.702 + 1.21i)11-s + (−0.0911 + 0.517i)12-s + (−0.210 − 0.0766i)13-s + (0.0775 + 0.134i)14-s + (0.703 + 0.590i)15-s + (0.191 + 0.160i)16-s + (0.565 − 0.205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.373831 + 0.232847i\)
\(L(\frac12)\) \(\approx\) \(0.373831 + 0.232847i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.694 + 3.93i)T \)
37 \( 1 + (6.74e3 - 4.88e3i)T \)
good3 \( 1 + (2.84 + 16.1i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (37.4 - 31.4i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (21.7 - 18.2i)T + (2.91e3 - 1.65e4i)T^{2} \)
11 \( 1 + (281. - 487. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (128. + 46.7i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (-673. + 245. i)T + (1.08e6 - 9.12e5i)T^{2} \)
19 \( 1 + (166. + 943. i)T + (-2.32e6 + 8.46e5i)T^{2} \)
23 \( 1 + (-1.45e3 - 2.51e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (4.24e3 - 7.35e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 2.02e3T + 2.86e7T^{2} \)
41 \( 1 + (3.26e3 + 1.18e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + 3.86e3T + 1.47e8T^{2} \)
47 \( 1 + (6.98e3 + 1.21e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (6.09e3 + 5.11e3i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (5.75e3 + 4.82e3i)T + (1.24e8 + 7.04e8i)T^{2} \)
61 \( 1 + (4.10e4 + 1.49e4i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (984. - 825. i)T + (2.34e8 - 1.32e9i)T^{2} \)
71 \( 1 + (-2.17e3 - 1.23e4i)T + (-1.69e9 + 6.17e8i)T^{2} \)
73 \( 1 + 7.40e3T + 2.07e9T^{2} \)
79 \( 1 + (-8.18e3 + 6.87e3i)T + (5.34e8 - 3.03e9i)T^{2} \)
83 \( 1 + (-6.05e4 + 2.20e4i)T + (3.01e9 - 2.53e9i)T^{2} \)
89 \( 1 + (1.40e4 + 1.17e4i)T + (9.69e8 + 5.49e9i)T^{2} \)
97 \( 1 + (-4.76e4 - 8.25e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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

−13.35604693756098200341611810122, −12.60717448035665100529518635425, −11.79114433762312929656732527091, −10.68130945577193903387659353029, −9.424489287741813590989883876248, −7.66371957055384884518791618935, −6.96526217101211039085017505866, −5.10604745626975919744201729256, −3.27526252618747049527229244791, −1.69198638401707796276777738792, 0.18997790836847177773618499255, 3.59301002808005022155404671505, 4.71107106113341713327201175616, 5.89144854153775110845451141073, 7.68470512119470363684900545068, 8.672182005008169521757063869873, 9.956302308165952547977041719902, 11.02551562171214743286636230714, 12.40506243334842110329583118245, 13.51678161103803596797136420202

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