Properties

Label 2-525-105.44-c0-0-0
Degree $2$
Conductor $525$
Sign $0.497 - 0.867i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 + 0.499i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 + 0.499i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ 0.497 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8144140001\)
\(L(\frac12)\) \(\approx\) \(0.8144140001\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.29086795088706991558262623613, −10.61662149592261850202692796156, −9.563383376064089929864771658994, −8.485526709763326895231228389587, −7.52305118607046320499038417515, −6.71585031807982118020629669133, −5.67969635074650773342840217999, −4.37968090057031358885415286030, −3.80054616729598062928015936041, −1.94263208975749447212563756240, 1.31700970631783550491257665869, 2.57763652978907088229204792629, 4.72499988851361415917537063770, 5.40749832722790060450139866586, 6.21648645733122950963363322555, 7.17021858082527622158894999152, 8.070664119349607603211823988416, 9.239248097409725813780480092263, 10.44283563240889743046930479483, 10.94112423295165005585658954793

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