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Properties

Label 2-525-105.59-c1-0-21
Degree $2$
Conductor $525$
Sign $0.999 + 0.00342i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)3-s + (1 − 1.73i)4-s + (2.59 + 0.5i)7-s + (−1.5 − 2.59i)9-s + (1.73 + 3i)12-s + 1.73·13-s + (−1.99 − 3.46i)16-s + (4.5 − 2.59i)19-s + (−3 + 3.46i)21-s + 5.19·27-s + (3.46 − 4i)28-s + (7.5 + 4.33i)31-s − 6·36-s + (0.866 − 0.5i)37-s + (−1.49 + 2.59i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.5 − 0.866i)4-s + (0.981 + 0.188i)7-s + (−0.5 − 0.866i)9-s + (0.499 + 0.866i)12-s + 0.480·13-s + (−0.499 − 0.866i)16-s + (1.03 − 0.596i)19-s + (−0.654 + 0.755i)21-s + 1.00·27-s + (0.654 − 0.755i)28-s + (1.34 + 0.777i)31-s − 36-s + (0.142 − 0.0821i)37-s + (−0.240 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.00342i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.999 + 0.00342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53136 - 0.00262172i\)
\(L(\frac12)\) \(\approx\) \(1.53136 - 0.00262172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 - 0.5i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (7.79 - 13.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−10.90764610772216959482419230022, −10.11237690998048633822049676863, −9.305465703520495916335120442305, −8.334128196953515337409331730175, −7.05004131096369171745656345858, −6.02559836105645710798297082611, −5.22579021989840989574310707739, −4.46143163244988302526808960807, −2.89295035940220094947687835329, −1.20034613732962439892339181137, 1.40341778610520331458527131625, 2.68398833672346315320723458304, 4.12151782406269923758295091544, 5.39145940801190247923921550748, 6.40382480122096659818858590081, 7.39014897151905333277519359354, 7.947493341785893684548648381964, 8.706429250091464032696820059905, 10.25156281897306566609144917516, 11.19736026962199556801713414618

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