Properties

Label 2-85176-1.1-c1-0-10
Degree $2$
Conductor $85176$
Sign $1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 7-s − 4·11-s − 6·17-s − 4·19-s − 6·23-s + 11·25-s − 4·35-s + 6·37-s + 12·41-s + 4·43-s − 6·47-s + 49-s + 8·53-s − 16·55-s − 14·59-s − 14·61-s − 4·67-s − 12·71-s − 6·73-s + 4·77-s + 8·79-s − 6·83-s − 24·85-s + 16·89-s − 16·95-s − 10·97-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 1.20·11-s − 1.45·17-s − 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.676·35-s + 0.986·37-s + 1.87·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 1.09·53-s − 2.15·55-s − 1.82·59-s − 1.79·61-s − 0.488·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 0.658·83-s − 2.60·85-s + 1.69·89-s − 1.64·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.735139345\)
\(L(\frac12)\) \(\approx\) \(1.735139345\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−13.78285940758175, −13.35257121533107, −13.14233534750179, −12.59927564378789, −12.16472600978183, −11.24891289715468, −10.67243365884662, −10.57157601741666, −9.919410763975069, −9.439877486924205, −9.023954147176296, −8.548889167207318, −7.765974013010443, −7.375268206808145, −6.403525767046846, −6.250879873474223, −5.831032089600887, −5.203294062519908, −4.475788375374662, −4.176828520372449, −2.930964621242341, −2.620626400109563, −2.077432899483368, −1.521322357050926, −0.3904913474755661, 0.3904913474755661, 1.521322357050926, 2.077432899483368, 2.620626400109563, 2.930964621242341, 4.176828520372449, 4.475788375374662, 5.203294062519908, 5.831032089600887, 6.250879873474223, 6.403525767046846, 7.375268206808145, 7.765974013010443, 8.548889167207318, 9.023954147176296, 9.439877486924205, 9.919410763975069, 10.57157601741666, 10.67243365884662, 11.24891289715468, 12.16472600978183, 12.59927564378789, 13.14233534750179, 13.35257121533107, 13.78285940758175

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