↑

Properties

Label 2-525-105.83-c0-0-0
Degree $2$
Conductor $525$
Sign $0.525 + 0.850i$
Analytic cond. $0.262009$
Root an. cond. $0.511868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s − i·4-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (0.707 + 0.707i)12-s + (1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)28-s − 1.00·36-s + 2.00i·39-s + (0.707 − 0.707i)48-s + 1.00i·49-s + (−1.41 − 1.41i)52-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − i·4-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (0.707 + 0.707i)12-s + (1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)28-s − 1.00·36-s + 2.00i·39-s + (0.707 − 0.707i)48-s + 1.00i·49-s + (−1.41 − 1.41i)52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6543182725\)
\(L(\frac12)\) \(\approx\) \(0.6543182725\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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

−10.73663685267931367003103823467, −10.22920659118094061618360877530, −9.502192133658940351846458793096, −8.453411896511049923328250370359, −7.00019859711888635160866455097, −6.08309271567896778719410962535, −5.51366751664060684876319640383, −4.30616079566097476951755520252, −3.26466027988382726509708545234, −0.940701655272083361873431912514, 1.98391067092541711650783629889, 3.35958962699595555300840156366, 4.56523487769665577352017974335, 5.99587655538827132802948794313, 6.58624572318556803402650697783, 7.50533220261942386304506457453, 8.584603722813116352980169948887, 9.170666523822853250203487380715, 10.56646222135400274494556508804, 11.64834201752974680973116365622

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