Properties

Label 2-2496-1.1-c1-0-37
Degree $2$
Conductor $2496$
Sign $-1$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 2·7-s + 9-s − 2·11-s + 13-s − 2·15-s + 6·17-s + 2·19-s − 2·21-s − 25-s + 27-s − 2·29-s + 2·31-s − 2·33-s + 4·35-s − 10·37-s + 39-s + 2·41-s − 8·43-s − 2·45-s − 2·47-s − 3·49-s + 6·51-s + 2·53-s + 4·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.516·15-s + 1.45·17-s + 0.458·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.676·35-s − 1.64·37-s + 0.160·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2496} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−8.472291229729577561267446968616, −7.70238653866752874546628860503, −7.32185771162981256016187886050, −6.25627621415601579990688034936, −5.39011374636217047998977702840, −4.39643067263263789920556453539, −3.34438303780590532331400946205, −3.10707452594267652556920073153, −1.55357312587466315401743793249, 0, 1.55357312587466315401743793249, 3.10707452594267652556920073153, 3.34438303780590532331400946205, 4.39643067263263789920556453539, 5.39011374636217047998977702840, 6.25627621415601579990688034936, 7.32185771162981256016187886050, 7.70238653866752874546628860503, 8.472291229729577561267446968616

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