Properties

Label 2-85176-1.1-c1-0-12
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s + 5·11-s − 7·17-s + 3·19-s − 3·23-s + 4·25-s − 7·29-s − 4·31-s + 3·35-s + 11·37-s + 4·41-s + 43-s + 8·47-s + 49-s − 2·53-s − 15·55-s + 6·59-s + 13·61-s + 4·67-s − 6·71-s + 73-s − 5·77-s + 8·79-s + 14·83-s + 21·85-s − 9·95-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 1.50·11-s − 1.69·17-s + 0.688·19-s − 0.625·23-s + 4/5·25-s − 1.29·29-s − 0.718·31-s + 0.507·35-s + 1.80·37-s + 0.624·41-s + 0.152·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 2.02·55-s + 0.781·59-s + 1.66·61-s + 0.488·67-s − 0.712·71-s + 0.117·73-s − 0.569·77-s + 0.900·79-s + 1.53·83-s + 2.27·85-s − 0.923·95-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

\(L(1)\) \(\approx\) \(1.278475017\)
\(L(\frac12)\) \(\approx\) \(1.278475017\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.89430382604813, −13.43488886620140, −12.84955927790637, −12.42678090189525, −11.82651868843668, −11.44115699137244, −11.16206967669830, −10.66124639141415, −9.697107429514672, −9.407626799618010, −8.941002449376648, −8.408756572760755, −7.760885703133745, −7.332172728962545, −6.840095531215241, −6.310650905425697, −5.773131007965288, −4.979922444000908, −4.234503746746318, −3.875907955980250, −3.682782147519573, −2.650267714085159, −2.083453426552551, −1.116667801896625, −0.4102308447458423, 0.4102308447458423, 1.116667801896625, 2.083453426552551, 2.650267714085159, 3.682782147519573, 3.875907955980250, 4.234503746746318, 4.979922444000908, 5.773131007965288, 6.310650905425697, 6.840095531215241, 7.332172728962545, 7.760885703133745, 8.408756572760755, 8.941002449376648, 9.407626799618010, 9.697107429514672, 10.66124639141415, 11.16206967669830, 11.44115699137244, 11.82651868843668, 12.42678090189525, 12.84955927790637, 13.43488886620140, 13.89430382604813

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