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Properties

Label 2-525-35.9-c1-0-23
Degree $2$
Conductor $525$
Sign $-0.652 + 0.758i$
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 − 0.5i)3-s + (−1 − 1.73i)4-s + (−0.866 − 2.5i)7-s + (0.499 − 0.866i)9-s + (−1.73 − 0.999i)12-s − i·13-s + (−1.99 + 3.46i)16-s + (−5.19 + 3i)17-s + (2.5 − 4.33i)19-s + (−2 − 1.73i)21-s + (−5.19 − 3i)23-s − 0.999i·27-s + (−3.46 + 4i)28-s + 6·29-s + (−2.5 − 4.33i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.5 − 0.866i)4-s + (−0.327 − 0.944i)7-s + (0.166 − 0.288i)9-s + (−0.499 − 0.288i)12-s − 0.277i·13-s + (−0.499 + 0.866i)16-s + (−1.26 + 0.727i)17-s + (0.573 − 0.993i)19-s + (−0.436 − 0.377i)21-s + (−1.08 − 0.625i)23-s − 0.192i·27-s + (−0.654 + 0.755i)28-s + 1.11·29-s + (−0.449 − 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.652 + 0.758i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.652 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484509 - 1.05575i\)
\(L(\frac12)\) \(\approx\) \(0.484509 - 1.05575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (5.19 - 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-9.52 + 5.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.52137372794257843106518252277, −9.610671163643757888769156915141, −8.888601298627591209746735380026, −7.88781663101204212672861021110, −6.81978965370785655899994689513, −6.05088864523935925593539434945, −4.68336931999107110531549574532, −3.85570900508396731422554963367, −2.24511596005071443987992856487, −0.64195063720172979189651612427, 2.31623614856788694332784806084, 3.36061847049378953327760695490, 4.37514452189771802736098331047, 5.46322416163517631314936764119, 6.75103492803242852751327424640, 7.81041586513481574412097072979, 8.637568460485584428216452568169, 9.244859779789085552063037310514, 10.01510899442284804488004593375, 11.33633197404245149494557869364

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