↑

Properties

Label 2-450-15.2-c3-0-5
Degree $2$
Conductor $450$
Sign $0.374 - 0.927i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)2-s − 4.00i·4-s + (23.9 + 23.9i)7-s + (−5.65 − 5.65i)8-s + 0.123i·11-s + (−58.9 + 58.9i)13-s + 67.6·14-s − 16.0·16-s + (−59.2 + 59.2i)17-s + 12.1i·19-s + (0.174 + 0.174i)22-s + (−19.7 − 19.7i)23-s + 166. i·26-s + (95.6 − 95.6i)28-s − 139.·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (1.29 + 1.29i)7-s + (−0.250 − 0.250i)8-s + 0.00337i·11-s + (−1.25 + 1.25i)13-s + 1.29·14-s − 0.250·16-s + (−0.845 + 0.845i)17-s + 0.146i·19-s + (0.00168 + 0.00168i)22-s + (−0.179 − 0.179i)23-s + 1.25i·26-s + (0.645 − 0.645i)28-s − 0.892·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.044079338\)
\(L(\frac12)\) \(\approx\) \(2.044079338\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-23.9 - 23.9i)T + 343iT^{2} \)
11 \( 1 - 0.123iT - 1.33e3T^{2} \)
13 \( 1 + (58.9 - 58.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (59.2 - 59.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 12.1iT - 6.85e3T^{2} \)
23 \( 1 + (19.7 + 19.7i)T + 1.21e4iT^{2} \)
29 \( 1 + 139.T + 2.43e4T^{2} \)
31 \( 1 + 71.8T + 2.97e4T^{2} \)
37 \( 1 + (-117. - 117. i)T + 5.06e4iT^{2} \)
41 \( 1 - 196. iT - 6.89e4T^{2} \)
43 \( 1 + (195. - 195. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-397. + 397. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-129. - 129. i)T + 1.48e5iT^{2} \)
59 \( 1 - 716.T + 2.05e5T^{2} \)
61 \( 1 - 114.T + 2.26e5T^{2} \)
67 \( 1 + (-375. - 375. i)T + 3.00e5iT^{2} \)
71 \( 1 - 232. iT - 3.57e5T^{2} \)
73 \( 1 + (-37.7 + 37.7i)T - 3.89e5iT^{2} \)
79 \( 1 - 57.2iT - 4.93e5T^{2} \)
83 \( 1 + (783. + 783. i)T + 5.71e5iT^{2} \)
89 \( 1 - 778.T + 7.04e5T^{2} \)
97 \( 1 + (887. + 887. i)T + 9.12e5iT^{2} \)
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

−11.23425355577846141498718348020, −10.00491269844023557824886266850, −9.055173631977126694929328555622, −8.313598516859251896227274781167, −7.06507810625490398960833230809, −5.87584668095056107263381064114, −4.96191533130129717800531703530, −4.16042999849983676951755950111, −2.41919099308308523190891523370, −1.78362309408834218002547264097, 0.52610008847972157544612276150, 2.31043209208487011301770007458, 3.83344522194509204905385567687, 4.79788840117333397898233812308, 5.49531730714138258182056940434, 7.11802244659872205720081938594, 7.47206220046400195629893723939, 8.378917892371064243383497145452, 9.639729810622362723656994523024, 10.68794222426048467120322157618

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