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Properties

Label 2-525-21.5-c1-0-28
Degree $2$
Conductor $525$
Sign $-0.997 + 0.0633i$
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  + (−1.5 − 0.866i)2-s − 1.73i·3-s + (0.5 + 0.866i)4-s + (−1.49 + 2.59i)6-s + (2.5 + 0.866i)7-s + 1.73i·8-s − 2.99·9-s + (3 − 1.73i)11-s + (1.50 − 0.866i)12-s − 3.46i·13-s + (−3 − 3.46i)14-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + (4.49 + 2.59i)18-s + (−6 − 3.46i)19-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s − 0.999i·3-s + (0.250 + 0.433i)4-s + (−0.612 + 1.06i)6-s + (0.944 + 0.327i)7-s + 0.612i·8-s − 0.999·9-s + (0.904 − 0.522i)11-s + (0.433 − 0.249i)12-s − 0.960i·13-s + (−0.801 − 0.925i)14-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + (1.06 + 0.612i)18-s + (−1.37 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.997 + 0.0633i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.997 + 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0219586 - 0.692742i\)
\(L(\frac12)\) \(\approx\) \(0.0219586 - 0.692742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6 + 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
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 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.72488767366802184456444912170, −9.181332958396641308130262522563, −8.810968646450715539072660421692, −7.975471045820858145695861668465, −7.07826082841924579101631793550, −5.89620916215843162358149701948, −4.83966341408845863366616686658, −2.87272187373347892593797480937, −1.85645975410394283444172087028, −0.59211553365999991701221482196, 1.77679179106632974928780606415, 4.04685548189866241618608465926, 4.34381341930807007258902733836, 6.02477940787102558058016703567, 6.84651061662732352503594710611, 8.046579033712958228665655895707, 8.676847126394447138055225033601, 9.328328095508950100191348210440, 10.27271836141236963311890415781, 10.90804375699040920097146219675

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