↑

Properties

Label 2-74-37.26-c7-0-4
Degree $2$
Conductor $74$
Sign $-0.727 + 0.685i$
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 + (1.38 − 2.39i)3-s + (−31.9 + 55.4i)4-s + (−125. + 216. i)5-s + 22.1·6-s + (133. − 231. i)7-s − 511.·8-s + (1.08e3 + 1.88e3i)9-s − 2.00e3·10-s − 2.17e3·11-s + (88.5 + 153. i)12-s + (−2.08e3 + 3.60e3i)13-s + 2.13e3·14-s + (345. + 598. i)15-s + (−2.04e3 − 3.54e3i)16-s + (−1.80e4 − 3.12e4i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.0295 − 0.0512i)3-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.0418·6-s + (0.147 − 0.254i)7-s − 0.353·8-s + (0.498 + 0.862i)9-s − 0.632·10-s − 0.492·11-s + (0.0147 + 0.0256i)12-s + (−0.262 + 0.454i)13-s + 0.207·14-s + (0.0264 + 0.0458i)15-s + (−0.125 − 0.216i)16-s + (−0.889 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.727 + 0.685i$
Analytic conductor: \(23.1164\)
Root analytic conductor: \(4.80796\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :7/2),\ -0.727 + 0.685i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.199275 - 0.502019i\)
\(L(\frac12)\) \(\approx\) \(0.199275 - 0.502019i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-1.90e5 + 2.41e5i)T \)
good3 \( 1 + (-1.38 + 2.39i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (125. - 216. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-133. + 231. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 2.17e3T + 1.94e7T^{2} \)
13 \( 1 + (2.08e3 - 3.60e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (1.80e4 + 3.12e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-2.33e3 + 4.04e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 4.00e4T + 3.40e9T^{2} \)
29 \( 1 + 8.54e4T + 1.72e10T^{2} \)
31 \( 1 + 3.19e5T + 2.75e10T^{2} \)
41 \( 1 + (-3.95e5 + 6.85e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 7.88e5T + 2.71e11T^{2} \)
47 \( 1 - 1.31e5T + 5.06e11T^{2} \)
53 \( 1 + (4.24e5 + 7.35e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-1.34e6 - 2.33e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.10e6 - 1.92e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (8.67e5 - 1.50e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (1.24e6 - 2.15e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 1.24e6T + 1.10e13T^{2} \)
79 \( 1 + (-1.61e5 + 2.79e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.84e6 - 6.65e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (-2.91e6 - 5.04e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 3.96e6T + 8.07e13T^{2} \)
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

−13.88405187321213974318180652866, −13.01528228020440442727731596539, −11.55950908245911827538953259127, −10.63802223163625687621051272287, −9.145631124005076270975637464916, −7.50294409241512277093089617711, −7.12657980350889957547968966009, −5.33600003721008846113635268655, −4.06689913531336768521233059857, −2.40371313111655631005759246952, 0.15657865428901898624854667646, 1.73887517844640876693006621218, 3.60407246025310478302577254893, 4.73758940595289300230282958458, 6.17034460147888835756436761120, 7.980899548586941063270744310339, 9.102069841257803144709140885167, 10.29161988789305595076531778872, 11.50076348956810480159564326401, 12.64729452655201518970918675233

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