Properties

Label 2-5e2-25.16-c3-0-2
Degree $2$
Conductor $25$
Sign $0.777 - 0.629i$
Analytic cond. $1.47504$
Root an. cond. $1.21451$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 + 1.65i)2-s + (0.411 + 1.26i)3-s + (−0.0362 − 0.111i)4-s + (9.59 + 5.74i)5-s + (−1.15 + 3.55i)6-s − 35.1·7-s + (7.04 − 21.6i)8-s + (20.4 − 14.8i)9-s + (12.3 + 28.8i)10-s + (−14.3 − 10.4i)11-s + (0.126 − 0.0916i)12-s + (0.739 − 0.537i)13-s + (−79.9 − 58.0i)14-s + (−3.32 + 14.5i)15-s + (51.0 − 37.0i)16-s + (−28.4 + 87.4i)17-s + ⋯
L(s)  = 1  + (0.803 + 0.583i)2-s + (0.0791 + 0.243i)3-s + (−0.00452 − 0.0139i)4-s + (0.858 + 0.513i)5-s + (−0.0785 + 0.241i)6-s − 1.89·7-s + (0.311 − 0.957i)8-s + (0.755 − 0.549i)9-s + (0.389 + 0.913i)10-s + (−0.392 − 0.285i)11-s + (0.00303 − 0.00220i)12-s + (0.0157 − 0.0114i)13-s + (−1.52 − 1.10i)14-s + (−0.0571 + 0.249i)15-s + (0.796 − 0.579i)16-s + (−0.405 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.777 - 0.629i$
Analytic conductor: \(1.47504\)
Root analytic conductor: \(1.21451\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :3/2),\ 0.777 - 0.629i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.53015 + 0.542033i\)
\(L(\frac12)\) \(\approx\) \(1.53015 + 0.542033i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.59 - 5.74i)T \)
good2 \( 1 + (-2.27 - 1.65i)T + (2.47 + 7.60i)T^{2} \)
3 \( 1 + (-0.411 - 1.26i)T + (-21.8 + 15.8i)T^{2} \)
7 \( 1 + 35.1T + 343T^{2} \)
11 \( 1 + (14.3 + 10.4i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (-0.739 + 0.537i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (28.4 - 87.4i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (19.0 - 58.5i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (-86.9 - 63.2i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (34.5 + 106. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-61.0 + 187. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-26.0 + 18.9i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (119. - 87.1i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 162.T + 7.95e4T^{2} \)
47 \( 1 + (8.22 + 25.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (159. + 490. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-91.9 + 66.7i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-162. - 117. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (102. - 316. i)T + (-2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (99.3 + 305. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-252. - 183. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-233. - 718. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-113. + 350. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (711. + 517. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (-208. - 640. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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

−16.86471697480839717318048529873, −15.68383180832849192216050750132, −14.89389032444020529263561880495, −13.32233488989567435353264630327, −12.90487919324501681936064458644, −10.25898528035548089725903060115, −9.578688543941371911286973009039, −6.74351284267614931627579452163, −5.94374587611115981413447646610, −3.63912519252289960702667370867, 2.75327784351150608070748842052, 4.91225543777962332635876644139, 6.85953181838970927795306165943, 9.110541905745112787116081130655, 10.42790112914620244270898812912, 12.43944994890092425491474768165, 13.09644688122295824575742757456, 13.74949436213358529957366484331, 15.84168893005331366080464394644, 16.82219424193151908192588591838

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