Properties

Label 2-62400-1.1-c1-0-179
Degree $2$
Conductor $62400$
Sign $-1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 13-s + 6·17-s − 4·23-s + 27-s + 10·29-s − 6·37-s − 39-s + 2·41-s + 4·43-s − 7·49-s + 6·51-s − 6·53-s − 6·61-s − 4·67-s − 4·69-s − 16·71-s + 2·73-s + 81-s − 4·83-s + 10·87-s − 6·89-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.277·13-s + 1.45·17-s − 0.834·23-s + 0.192·27-s + 1.85·29-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s − 49-s + 0.840·51-s − 0.824·53-s − 0.768·61-s − 0.488·67-s − 0.481·69-s − 1.89·71-s + 0.234·73-s + 1/9·81-s − 0.439·83-s + 1.07·87-s − 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−14.31981014773714, −14.16286899373255, −13.69037445469741, −12.95848523517599, −12.50767528707821, −12.00465804848204, −11.72239360411024, −10.79628805831192, −10.37802916818662, −9.916346049460439, −9.468550168774976, −8.838060153934522, −8.225702267475462, −7.922342150940020, −7.310363328743001, −6.770350918819871, −6.058857991389718, −5.622469427368493, −4.794552725857135, −4.415653780984523, −3.594088528655664, −3.080593505432527, −2.558833430825796, −1.657776372107243, −1.103184670806780, 0, 1.103184670806780, 1.657776372107243, 2.558833430825796, 3.080593505432527, 3.594088528655664, 4.415653780984523, 4.794552725857135, 5.622469427368493, 6.058857991389718, 6.770350918819871, 7.310363328743001, 7.922342150940020, 8.225702267475462, 8.838060153934522, 9.468550168774976, 9.916346049460439, 10.37802916818662, 10.79628805831192, 11.72239360411024, 12.00465804848204, 12.50767528707821, 12.95848523517599, 13.69037445469741, 14.16286899373255, 14.31981014773714

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