Properties

Label 2-168e2-1.1-c1-0-104
Degree $2$
Conductor $28224$
Sign $-1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·11-s + 13-s − 2·17-s + 5·19-s − 6·23-s − 5·25-s − 8·29-s + 3·31-s + 9·37-s + 2·41-s − 43-s + 8·47-s + 6·53-s − 6·59-s − 2·61-s + 5·67-s − 4·71-s + 11·73-s − 5·79-s + 12·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.603·11-s + 0.277·13-s − 0.485·17-s + 1.14·19-s − 1.25·23-s − 25-s − 1.48·29-s + 0.538·31-s + 1.47·37-s + 0.312·41-s − 0.152·43-s + 1.16·47-s + 0.824·53-s − 0.781·59-s − 0.256·61-s + 0.610·67-s − 0.474·71-s + 1.28·73-s − 0.562·79-s + 1.27·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{28224} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−15.38469631236815, −15.15760552075459, −14.31762030486783, −13.80965204199243, −13.43365890827612, −12.90862233347210, −12.20675754684038, −11.74542602055274, −11.18829018711238, −10.71370927591664, −9.939812047147349, −9.610329167760897, −9.017125924918145, −8.256684248650096, −7.706934844384785, −7.406543134094761, −6.516888443364294, −5.817717697658387, −5.559506215333594, −4.648529210971910, −4.028617870463217, −3.447745190743604, −2.540717693988303, −2.004568684089794, −0.9978249468049168, 0, 0.9978249468049168, 2.004568684089794, 2.540717693988303, 3.447745190743604, 4.028617870463217, 4.648529210971910, 5.559506215333594, 5.817717697658387, 6.516888443364294, 7.406543134094761, 7.706934844384785, 8.256684248650096, 9.017125924918145, 9.610329167760897, 9.939812047147349, 10.71370927591664, 11.18829018711238, 11.74542602055274, 12.20675754684038, 12.90862233347210, 13.43365890827612, 13.80965204199243, 14.31762030486783, 15.15760552075459, 15.38469631236815

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