↑

Properties

Label 2-1450-145.12-c1-0-1
Degree $2$
Conductor $1450$
Sign $-0.374 - 0.927i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  − i·2-s + 1.48·3-s − 4-s − 1.48i·6-s + (−2.18 − 2.18i)7-s + i·8-s − 0.783·9-s + (−1.48 − 1.48i)11-s − 1.48·12-s + (−0.792 − 0.792i)13-s + (−2.18 + 2.18i)14-s + 16-s + 7.13i·17-s + 0.783i·18-s + (−5.65 + 5.65i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.859·3-s − 0.5·4-s − 0.607i·6-s + (−0.824 − 0.824i)7-s + 0.353i·8-s − 0.261·9-s + (−0.448 − 0.448i)11-s − 0.429·12-s + (−0.219 − 0.219i)13-s + (−0.582 + 0.582i)14-s + 0.250·16-s + 1.73i·17-s + 0.184i·18-s + (−1.29 + 1.29i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1088262581\)
\(L(\frac12)\) \(\approx\) \(0.1088262581\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
29 \( 1 + (3.65 + 3.95i)T \)
good3 \( 1 - 1.48T + 3T^{2} \)
7 \( 1 + (2.18 + 2.18i)T + 7iT^{2} \)
11 \( 1 + (1.48 + 1.48i)T + 11iT^{2} \)
13 \( 1 + (0.792 + 0.792i)T + 13iT^{2} \)
17 \( 1 - 7.13iT - 17T^{2} \)
19 \( 1 + (5.65 - 5.65i)T - 19iT^{2} \)
23 \( 1 + (0.207 - 0.207i)T - 23iT^{2} \)
31 \( 1 + (-2.76 - 2.76i)T + 31iT^{2} \)
37 \( 1 - 0.0177T + 37T^{2} \)
41 \( 1 + (3.56 - 3.56i)T - 41iT^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 - 3.96T + 47T^{2} \)
53 \( 1 + (5.94 - 5.94i)T - 53iT^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 + (6.05 + 6.05i)T + 61iT^{2} \)
67 \( 1 + (10.2 - 10.2i)T - 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + 1.18iT - 73T^{2} \)
79 \( 1 + (-4.07 + 4.07i)T - 79iT^{2} \)
83 \( 1 + (7.70 - 7.70i)T - 83iT^{2} \)
89 \( 1 + (1.06 - 1.06i)T - 89iT^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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

−9.858618061119390088877948084625, −9.020054952455533658823380198832, −8.169407738160509033426952051131, −7.79756474655228499207200052344, −6.38243180643892115558883876416, −5.75078772911351882321492346783, −4.21230874268346450826725161654, −3.66663871081978168989809821068, −2.80981085470612914703134458741, −1.71431135981699485675116629589, 0.03572234111187658924741191233, 2.43669326937800026373722888379, 2.89573082560726474461921825431, 4.24452540317844357333271538538, 5.18332791983911691707351503039, 6.03972123094272704042605908368, 6.99698215509673569604856926668, 7.57202645539208288193522636601, 8.649252256463479819877798612201, 9.145017536761046140533883749308

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