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Properties

Label 2-525-21.17-c1-0-17
Degree $2$
Conductor $525$
Sign $0.205 - 0.978i$
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)3-s + (−1 + 1.73i)4-s + (2 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−3 + 1.73i)12-s + 6.92i·13-s + (−1.99 − 3.46i)16-s + (3 − 1.73i)19-s + (4.5 − 0.866i)21-s + 5.19i·27-s + (0.999 + 5.19i)28-s + (−7.5 − 4.33i)31-s − 6·36-s + (5.5 + 9.52i)37-s + (−5.99 + 10.3i)39-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.755 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s + 1.92i·13-s + (−0.499 − 0.866i)16-s + (0.688 − 0.397i)19-s + (0.981 − 0.188i)21-s + 0.999i·27-s + (0.188 + 0.981i)28-s + (−1.34 − 0.777i)31-s − 36-s + (0.904 + 1.56i)37-s + (−0.960 + 1.66i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40753 + 1.14296i\)
\(L(\frac12)\) \(\approx\) \(1.40753 + 1.14296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
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 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
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 \( 1 + (-13.5 + 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−11.20898337722182738029236542287, −9.859324735411588998911630704885, −9.235219852338695996066870610755, −8.426284183633319152442091242206, −7.63114268332235906291337470677, −6.86686745065389677772433983252, −4.92459321086285908392015516086, −4.28750309905247473683710449697, −3.41195056851713465102309223673, −1.95392133223678978025031114901, 1.10319902927824971887898493440, 2.44495539396809546843751774069, 3.76773103021466577869378495194, 5.24684148744016270174242739152, 5.79601223912743460586308788474, 7.23859822324037744774027550113, 8.142644917649185827675823322083, 8.790112898646503023663470037703, 9.704529820792939241649061447679, 10.49831278620155977143497500687

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