↑

Properties

Label 2-525-15.2-c1-0-35
Degree $2$
Conductor $525$
Sign $-0.994 - 0.102i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 − 1.90i)2-s + (−0.394 − 1.68i)3-s − 5.22i·4-s + (−3.95 − 2.45i)6-s + (0.707 + 0.707i)7-s + (−6.13 − 6.13i)8-s + (−2.68 + 1.33i)9-s + 3.76i·11-s + (−8.81 + 2.06i)12-s + (3.48 − 3.48i)13-s + 2.68·14-s − 12.8·16-s + (−0.131 + 0.131i)17-s + (−2.57 + 7.64i)18-s − 3.89i·19-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)2-s + (−0.227 − 0.973i)3-s − 2.61i·4-s + (−1.61 − 1.00i)6-s + (0.267 + 0.267i)7-s + (−2.16 − 2.16i)8-s + (−0.896 + 0.443i)9-s + 1.13i·11-s + (−2.54 + 0.595i)12-s + (0.965 − 0.965i)13-s + 0.718·14-s − 3.21·16-s + (−0.0319 + 0.0319i)17-s + (−0.608 + 1.80i)18-s − 0.893i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.994 - 0.102i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.994 - 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133079 + 2.59190i\)
\(L(\frac12)\) \(\approx\) \(0.133079 + 2.59190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.394 + 1.68i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.90 + 1.90i)T - 2iT^{2} \)
11 \( 1 - 3.76iT - 11T^{2} \)
13 \( 1 + (-3.48 + 3.48i)T - 13iT^{2} \)
17 \( 1 + (0.131 - 0.131i)T - 17iT^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + (-3.35 - 3.35i)T + 23iT^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 + (-4.98 - 4.98i)T + 37iT^{2} \)
41 \( 1 + 1.16iT - 41T^{2} \)
43 \( 1 + (2.05 - 2.05i)T - 43iT^{2} \)
47 \( 1 + (-7.97 + 7.97i)T - 47iT^{2} \)
53 \( 1 + (3.80 + 3.80i)T + 53iT^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 + 4.11T + 61T^{2} \)
67 \( 1 + (0.153 + 0.153i)T + 67iT^{2} \)
71 \( 1 - 2.12iT - 71T^{2} \)
73 \( 1 + (9.79 - 9.79i)T - 73iT^{2} \)
79 \( 1 + 0.147iT - 79T^{2} \)
83 \( 1 + (-2.58 - 2.58i)T + 83iT^{2} \)
89 \( 1 - 1.17T + 89T^{2} \)
97 \( 1 + (-1.52 - 1.52i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88932875415224063389061808428, −9.932091920484137306951707717663, −8.809919171286052196487833565747, −7.44756354826243129539791501858, −6.34908697850439050308755168649, −5.48081053876870939856448978832, −4.67280799257380268003971580075, −3.28818096141480748290529193721, −2.27160072064971831202389634347, −1.14437285345343852852549791233, 3.12853054743060405173042176286, 4.00225032889183337383221401933, 4.70342157322962221268809221305, 5.90863921943858404454128215190, 6.21785805100487551317945530490, 7.53746335501784420542610099033, 8.515638570024143529565333943147, 9.144778379052657264032235301022, 10.76670515334238617610276491476, 11.39528600428313107343448372135

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