Properties

Degree $2$
Conductor $7350$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 5·11-s + 12-s + 13-s + 16-s − 2·17-s − 18-s + 7·19-s + 5·22-s + 3·23-s − 24-s − 26-s + 27-s − 6·31-s − 32-s − 5·33-s + 2·34-s + 36-s − 5·37-s − 7·38-s + 39-s − 9·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.60·19-s + 1.06·22-s + 0.625·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.07·31-s − 0.176·32-s − 0.870·33-s + 0.342·34-s + 1/6·36-s − 0.821·37-s − 1.13·38-s + 0.160·39-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{7350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−17.59249527684541, −16.80255883028417, −16.02268447234853, −15.82178935715345, −15.26144446657907, −14.49001543017108, −13.91651254419034, −13.13415167225695, −12.88905256430205, −11.94986654411659, −11.26700222356223, −10.73594732247753, −10.03517531157333, −9.577831799825400, −8.771880940246534, −8.361457810857654, −7.449886862184653, −7.321929308891440, −6.312084194387793, −5.360572977213088, −4.915341258457568, −3.626483750842182, −3.032438350122558, −2.239414337705550, −1.292019043526650, 0, 1.292019043526650, 2.239414337705550, 3.032438350122558, 3.626483750842182, 4.915341258457568, 5.360572977213088, 6.312084194387793, 7.321929308891440, 7.449886862184653, 8.361457810857654, 8.771880940246534, 9.577831799825400, 10.03517531157333, 10.73594732247753, 11.26700222356223, 11.94986654411659, 12.88905256430205, 13.13415167225695, 13.91651254419034, 14.49001543017108, 15.26144446657907, 15.82178935715345, 16.02268447234853, 16.80255883028417, 17.59249527684541

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