↑

Properties

Label 2-525-105.2-c1-0-21
Degree $2$
Conductor $525$
Sign $0.996 - 0.0838i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0799 + 0.298i)2-s + (−1.29 + 1.15i)3-s + (1.64 − 0.952i)4-s + (−0.447 − 0.292i)6-s + (2.46 − 0.951i)7-s + (0.852 + 0.852i)8-s + (0.333 − 2.98i)9-s + (−0.660 + 0.381i)11-s + (−1.02 + 3.13i)12-s + (2.27 − 2.27i)13-s + (0.481 + 0.660i)14-s + (1.71 − 2.97i)16-s + (−4.69 − 1.25i)17-s + (0.916 − 0.138i)18-s + (−1.41 − 0.818i)19-s + ⋯
L(s)  = 1  + (0.0565 + 0.210i)2-s + (−0.745 + 0.666i)3-s + (0.824 − 0.476i)4-s + (−0.182 − 0.119i)6-s + (0.933 − 0.359i)7-s + (0.301 + 0.301i)8-s + (0.111 − 0.993i)9-s + (−0.199 + 0.114i)11-s + (−0.297 + 0.904i)12-s + (0.629 − 0.629i)13-s + (0.128 + 0.176i)14-s + (0.429 − 0.744i)16-s + (−1.13 − 0.305i)17-s + (0.215 − 0.0327i)18-s + (−0.325 − 0.187i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.996 - 0.0838i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.996 - 0.0838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57429 + 0.0661101i\)
\(L(\frac12)\) \(\approx\) \(1.57429 + 0.0661101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 - 1.15i)T \)
5 \( 1 \)
7 \( 1 + (-2.46 + 0.951i)T \)
good2 \( 1 + (-0.0799 - 0.298i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.660 - 0.381i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 2.27i)T - 13iT^{2} \)
17 \( 1 + (4.69 + 1.25i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.41 + 0.818i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.39 + 1.98i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.41 - 0.915i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (2.69 - 2.69i)T - 43iT^{2} \)
47 \( 1 + (-1.10 - 4.14i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.79 - 6.71i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.84 + 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0126 - 0.0471i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-1.34 - 0.359i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.66 - 2.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.05 + 5.05i)T + 83iT^{2} \)
89 \( 1 + (-0.453 + 0.785i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.73 + 3.73i)T + 97iT^{2} \)
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

−10.76086743599922737495827726487, −10.48498920655764581084898131768, −9.172697038435376557938484540596, −8.157232652217991425570164515575, −6.95211743620355121480420176605, −6.31471377639351232920855224047, −5.12406973415847918590846547083, −4.58905680203103682996913523779, −2.92992381515636605370739247645, −1.18195596037706997568549533144, 1.51456938700592276217809606368, 2.51337266379937729705191419978, 4.18560656911718115423797643207, 5.32652621718665147050483844334, 6.43248460496001042708023428288, 7.06122379309242294987651177451, 8.097083929544640065695900984406, 8.796366780815314721348057896508, 10.45150261625533978709471581752, 11.14476134095881729193816990835

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