Properties

Label 2-525-7.6-c2-0-40
Degree $2$
Conductor $525$
Sign $0.936 - 0.349i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.80·2-s + 1.73i·3-s + 10.4·4-s + 6.58i·6-s + (−2.44 − 6.55i)7-s + 24.6·8-s − 2.99·9-s + 14.4·11-s + 18.1i·12-s + 16.9i·13-s + (−9.30 − 24.9i)14-s + 51.8·16-s + 13.0i·17-s − 11.4·18-s − 18.6i·19-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.577i·3-s + 2.61·4-s + 1.09i·6-s + (−0.349 − 0.936i)7-s + 3.07·8-s − 0.333·9-s + 1.31·11-s + 1.51i·12-s + 1.30i·13-s + (−0.664 − 1.78i)14-s + 3.23·16-s + 0.765i·17-s − 0.634·18-s − 0.983i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.936 - 0.349i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.819285135\)
\(L(\frac12)\) \(\approx\) \(5.819285135\)
\(L(2)\) 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 \( 1 + (2.44 + 6.55i)T \)
good2 \( 1 - 3.80T + 4T^{2} \)
11 \( 1 - 14.4T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 - 13.0iT - 289T^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
23 \( 1 + 10.3T + 529T^{2} \)
29 \( 1 + 13.7T + 841T^{2} \)
31 \( 1 + 42.4iT - 961T^{2} \)
37 \( 1 + 28.7T + 1.36e3T^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 - 5.84T + 1.84e3T^{2} \)
47 \( 1 - 10.5iT - 2.20e3T^{2} \)
53 \( 1 + 81.9T + 2.80e3T^{2} \)
59 \( 1 + 35.1iT - 3.48e3T^{2} \)
61 \( 1 - 68.4iT - 3.72e3T^{2} \)
67 \( 1 + 47.4T + 4.48e3T^{2} \)
71 \( 1 - 47.6T + 5.04e3T^{2} \)
73 \( 1 + 125. iT - 5.32e3T^{2} \)
79 \( 1 + 129.T + 6.24e3T^{2} \)
83 \( 1 + 42.6iT - 6.88e3T^{2} \)
89 \( 1 + 25.5iT - 7.92e3T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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

−11.20310468884322756558356998522, −10.10250556703072204376730409361, −9.102661787279524038560499716013, −7.50970656792414431499801579664, −6.60225652803154682195882544935, −6.06068064268588563425594701971, −4.59161992830263020721711124111, −4.14998588138403513711984210218, −3.32801829787824733695463310171, −1.78186320180543805269821874407, 1.66925088345384869660788195697, 2.92226128633836231710958390266, 3.70712190953444251130850666287, 5.11278091766268369694394155079, 5.82971900804358389660478249628, 6.55648807028828724850750458529, 7.45605711731066703663288671452, 8.615321173871753091900433552398, 9.974724356682058189977522824795, 11.11974016204071613482522778177

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