Properties

Degree $2$
Conductor $215475$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 4·11-s − 12-s − 16-s − 17-s − 18-s + 4·19-s + 4·22-s + 3·24-s + 27-s − 2·29-s + 8·31-s − 5·32-s − 4·33-s + 34-s − 36-s − 2·37-s − 4·38-s − 2·41-s + 4·43-s + 4·44-s + 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.852·22-s + 0.612·24-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.883·32-s − 0.696·33-s + 0.171·34-s − 1/6·36-s − 0.328·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(215475\)    =    \(3 \cdot 5^{2} \cdot 13^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{215475} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 215475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.590008258\)
\(L(\frac12)\) \(\approx\) \(1.590008258\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
13 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.19571967101416, −12.69469396727290, −11.99757074343033, −11.59584043621115, −10.94235496480005, −10.46192280885251, −10.00456338068076, −9.844443715601386, −9.150540082009580, −8.639100505445041, −8.469355693847805, −7.769964811863158, −7.481823301969229, −7.088029554304076, −6.338546672705837, −5.574141063551252, −5.293264616485618, −4.567182605614173, −4.267387376324853, −3.511609226074578, −2.959083551830470, −2.406521450947105, −1.772680934730984, −1.011267419018328, −0.4497718739118864, 0.4497718739118864, 1.011267419018328, 1.772680934730984, 2.406521450947105, 2.959083551830470, 3.511609226074578, 4.267387376324853, 4.567182605614173, 5.293264616485618, 5.574141063551252, 6.338546672705837, 7.088029554304076, 7.481823301969229, 7.769964811863158, 8.469355693847805, 8.639100505445041, 9.150540082009580, 9.844443715601386, 10.00456338068076, 10.46192280885251, 10.94235496480005, 11.59584043621115, 11.99757074343033, 12.69469396727290, 13.19571967101416

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