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Properties

Label 2-525-7.4-c1-0-16
Degree $2$
Conductor $525$
Sign $0.934 - 0.354i$
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  + (0.890 + 1.54i)2-s + (0.5 − 0.866i)3-s + (−0.587 + 1.01i)4-s + 1.78·6-s + (−1.09 − 2.40i)7-s + 1.47·8-s + (−0.499 − 0.866i)9-s + (1.03 − 1.80i)11-s + (0.587 + 1.01i)12-s + 3.13·13-s + (2.74 − 3.83i)14-s + (2.48 + 4.30i)16-s + (1.06 − 1.84i)17-s + (0.890 − 1.54i)18-s + (3.86 + 6.70i)19-s + ⋯
L(s)  = 1  + (0.629 + 1.09i)2-s + (0.288 − 0.499i)3-s + (−0.293 + 0.508i)4-s + 0.727·6-s + (−0.413 − 0.910i)7-s + 0.520·8-s + (−0.166 − 0.288i)9-s + (0.313 − 0.542i)11-s + (0.169 + 0.293i)12-s + 0.868·13-s + (0.732 − 1.02i)14-s + (0.621 + 1.07i)16-s + (0.258 − 0.447i)17-s + (0.209 − 0.363i)18-s + (0.887 + 1.53i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.934 - 0.354i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.934 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27392 + 0.417199i\)
\(L(\frac12)\) \(\approx\) \(2.27392 + 0.417199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.09 + 2.40i)T \)
good2 \( 1 + (-0.890 - 1.54i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.03 + 1.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.13T + 13T^{2} \)
17 \( 1 + (-1.06 + 1.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.86 - 6.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.76 + 4.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 + (1.45 - 2.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.75 + 3.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.99T + 41T^{2} \)
43 \( 1 + 4.99T + 43T^{2} \)
47 \( 1 + (1.22 + 2.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.47 - 2.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.44 - 9.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.91 - 3.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + (-4.27 + 7.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.05 + 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.75T + 83T^{2} \)
89 \( 1 + (0.309 + 0.535i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.296T + 97T^{2} \)
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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

−10.84919669646565798563322402211, −10.05944139718500877884651662233, −8.825176690424822850638936942097, −7.82859854475078856369817684688, −7.25420230928808220731684764877, −6.25456032301272090052968654210, −5.69611578340413443553002893872, −4.22819608980230215809774772215, −3.38973548134483066104670735497, −1.32742290648075050959299013481, 1.79252143607499724430314983314, 2.98939611218686956416736785949, 3.76535876625101225635977829982, 4.87707689460303672936614968673, 5.84136765003643842727180295483, 7.19733285972555808222008551013, 8.350767367752080609507510462724, 9.440790530169124648995917995952, 9.875322872805439286233284702792, 11.21568535972034040312979141651

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