Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 5·5-s + 7·7-s − 42·11-s + 20·13-s + 64·16-s − 66·17-s + 38·19-s + 40·20-s − 12·23-s + 25·25-s − 56·28-s + 258·29-s + 146·31-s − 35·35-s + 434·37-s + 282·41-s + 20·43-s + 336·44-s + 72·47-s + 49·49-s − 160·52-s − 336·53-s + 210·55-s + 360·59-s − 682·61-s − 512·64-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s + 0.377·7-s − 1.15·11-s + 0.426·13-s + 16-s − 0.941·17-s + 0.458·19-s + 0.447·20-s − 0.108·23-s + 1/5·25-s − 0.377·28-s + 1.65·29-s + 0.845·31-s − 0.169·35-s + 1.92·37-s + 1.07·41-s + 0.0709·43-s + 1.15·44-s + 0.223·47-s + 1/7·49-s − 0.426·52-s − 0.870·53-s + 0.514·55-s + 0.794·59-s − 1.43·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{315} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.160452099\)
\(L(\frac12)\)  \(\approx\)  \(1.160452099\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + p T \)
7 \( 1 - p T \)
good2 \( 1 + p^{3} T^{2} \)
11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
19 \( 1 - 2 p T + p^{3} T^{2} \)
23 \( 1 + 12 T + p^{3} T^{2} \)
29 \( 1 - 258 T + p^{3} T^{2} \)
31 \( 1 - 146 T + p^{3} T^{2} \)
37 \( 1 - 434 T + p^{3} T^{2} \)
41 \( 1 - 282 T + p^{3} T^{2} \)
43 \( 1 - 20 T + p^{3} T^{2} \)
47 \( 1 - 72 T + p^{3} T^{2} \)
53 \( 1 + 336 T + p^{3} T^{2} \)
59 \( 1 - 360 T + p^{3} T^{2} \)
61 \( 1 + 682 T + p^{3} T^{2} \)
67 \( 1 - 812 T + p^{3} T^{2} \)
71 \( 1 + 810 T + p^{3} T^{2} \)
73 \( 1 + 124 T + p^{3} T^{2} \)
79 \( 1 - 1136 T + p^{3} T^{2} \)
83 \( 1 + 156 T + p^{3} T^{2} \)
89 \( 1 - 1038 T + p^{3} T^{2} \)
97 \( 1 - 1208 T + p^{3} T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.15286715666112145620963228344, −10.31153961233342374182773609078, −9.280789191558558104943270885003, −8.295523579711364497311780079404, −7.72871184383678397496126096351, −6.21760747313520108846720695270, −4.97444269200799989663493631341, −4.24540220067398550921794667440, −2.78318453049259369358140902160, −0.74970079008515279609022174604, 0.74970079008515279609022174604, 2.78318453049259369358140902160, 4.24540220067398550921794667440, 4.97444269200799989663493631341, 6.21760747313520108846720695270, 7.72871184383678397496126096351, 8.295523579711364497311780079404, 9.280789191558558104943270885003, 10.31153961233342374182773609078, 11.15286715666112145620963228344

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