↑

Properties

Label 2-275-55.39-c2-0-0
Degree $2$
Conductor $275$
Sign $-0.418 + 0.908i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 1.80i)2-s + (−2.62 + 3.61i)3-s + (0.309 − 0.224i)4-s + (−8.09 − 2.62i)6-s + (−6.74 + 4.89i)7-s + (6.74 + 4.89i)8-s + (−3.39 − 10.4i)9-s + (−10.3 + 3.66i)11-s + 1.70i·12-s + (−6.06 − 18.6i)13-s + (−12.8 − 9.31i)14-s + (−4.42 + 13.6i)16-s + (−6.65 + 20.4i)17-s + (16.9 − 12.2i)18-s + (9.14 − 12.5i)19-s + ⋯
L(s)  = 1  + (0.293 + 0.904i)2-s + (−0.876 + 1.20i)3-s + (0.0772 − 0.0561i)4-s + (−1.34 − 0.438i)6-s + (−0.963 + 0.699i)7-s + (0.842 + 0.612i)8-s + (−0.377 − 1.16i)9-s + (−0.942 + 0.333i)11-s + 0.142i·12-s + (−0.466 − 1.43i)13-s + (−0.916 − 0.665i)14-s + (−0.276 + 0.851i)16-s + (−0.391 + 1.20i)17-s + (0.940 − 0.683i)18-s + (0.481 − 0.662i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.361727 - 0.564647i\)
\(L(\frac12)\) \(\approx\) \(0.361727 - 0.564647i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (10.3 - 3.66i)T \)
good2 \( 1 + (-0.587 - 1.80i)T + (-3.23 + 2.35i)T^{2} \)
3 \( 1 + (2.62 - 3.61i)T + (-2.78 - 8.55i)T^{2} \)
7 \( 1 + (6.74 - 4.89i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (6.06 + 18.6i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (6.65 - 20.4i)T + (-233. - 169. i)T^{2} \)
19 \( 1 + (-9.14 + 12.5i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 6.50iT - 529T^{2} \)
29 \( 1 + (2.70 + 3.71i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-9.93 - 30.5i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (26.5 + 36.5i)T + (-423. + 1.30e3i)T^{2} \)
41 \( 1 + (17.0 - 23.4i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 65.3T + 1.84e3T^{2} \)
47 \( 1 + (10.0 - 13.7i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (78.3 - 25.4i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-1.25 + 0.910i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (24.0 + 7.80i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 76.0iT - 4.48e3T^{2} \)
71 \( 1 + (33.6 - 103. i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (96.3 - 69.9i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (80.1 - 26.0i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (23.6 - 72.9i)T + (-5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 53.7T + 7.92e3T^{2} \)
97 \( 1 + (155. - 50.4i)T + (7.61e3 - 5.53e3i)T^{2} \)
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

−12.42703670559124910504168070560, −11.03719789029970890153661310285, −10.45688589700210195914275913115, −9.721840567884343623422840236461, −8.393661047705461964469298538832, −7.16890875785748380498899239818, −5.92903822538506722037485263327, −5.49635970707073416498441392453, −4.51613274543003430221784734864, −2.85189371528017166051824743756, 0.31927750867865369662204705707, 1.86312536850717559233566777174, 3.19590196706067878810485149211, 4.68202623642452747627364506360, 6.16674729586785054839945729944, 7.08127221399963013248717516481, 7.59352899723998307275941865908, 9.464532158701658220275323570403, 10.40878364106825812674877388976, 11.40156773542744116517094964834

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