↑

Properties

Label 2-85680-1.1-c1-0-125
Degree $2$
Conductor $85680$
Sign $-1$
Analytic cond. $684.158$
Root an. cond. $26.1564$
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  − 5-s + 7-s + 17-s + 6·19-s + 6·23-s + 25-s + 4·29-s − 4·31-s − 35-s − 4·37-s + 6·41-s − 12·43-s + 6·47-s + 49-s − 6·53-s − 12·59-s − 2·61-s − 4·67-s + 8·71-s + 6·73-s + 2·79-s − 10·83-s − 85-s − 16·89-s − 6·95-s + 18·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.718·31-s − 0.169·35-s − 0.657·37-s + 0.937·41-s − 1.82·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.225·79-s − 1.09·83-s − 0.108·85-s − 1.69·89-s − 0.615·95-s + 1.82·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(684.158\)
Root analytic conductor: \(26.1564\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85680,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−14.00789409498105, −13.90922338217972, −13.14073403515364, −12.59447587391315, −12.24553467190752, −11.59663023264099, −11.28705556969990, −10.78010572593917, −10.19456579089401, −9.697292994452656, −8.991842980572002, −8.805553277303637, −7.924190963341705, −7.683707664326750, −7.090084555918898, −6.606088848256356, −5.892047892838846, −5.220391078719334, −4.912925760946735, −4.279618745807248, −3.434412018212510, −3.174194306796371, −2.404461994543873, −1.472662090258720, −1.004272959473276, 0, 1.004272959473276, 1.472662090258720, 2.404461994543873, 3.174194306796371, 3.434412018212510, 4.279618745807248, 4.912925760946735, 5.220391078719334, 5.892047892838846, 6.606088848256356, 7.090084555918898, 7.683707664326750, 7.924190963341705, 8.805553277303637, 8.991842980572002, 9.697292994452656, 10.19456579089401, 10.78010572593917, 11.28705556969990, 11.59663023264099, 12.24553467190752, 12.59447587391315, 13.14073403515364, 13.90922338217972, 14.00789409498105

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