Properties

Degree $2$
Conductor $155526$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s + 4·22-s + 24-s − 25-s + 2·26-s + 27-s − 2·29-s − 2·30-s + 8·31-s + 32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.176·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155526\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{155526} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 155526,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.49828547101072, −13.27543956809174, −12.55852730901803, −11.99093373162077, −11.85616118172244, −11.28733039918371, −10.79158589223650, −10.40269118581424, −9.498494168262533, −9.196372901836392, −8.779492036713733, −8.092782527355175, −7.754115358580081, −7.116117096997110, −6.758461689082947, −6.170145277917747, −5.727490540762138, −4.858724112282533, −4.326683570642734, −4.071362231694874, −3.530390286258680, −2.951525980134916, −2.387436857593290, −1.560508731339247, −1.056983237660951, 0, 1.056983237660951, 1.560508731339247, 2.387436857593290, 2.951525980134916, 3.530390286258680, 4.071362231694874, 4.326683570642734, 4.858724112282533, 5.727490540762138, 6.170145277917747, 6.758461689082947, 7.116117096997110, 7.754115358580081, 8.092782527355175, 8.779492036713733, 9.196372901836392, 9.498494168262533, 10.40269118581424, 10.79158589223650, 11.28733039918371, 11.85616118172244, 11.99093373162077, 12.55852730901803, 13.27543956809174, 13.49828547101072

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