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Properties

Label 2-408e2-1.1-c1-0-65
Degree $2$
Conductor $166464$
Sign $1$
Analytic cond. $1329.22$
Root an. cond. $36.4584$
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  + 2·5-s + 4·11-s + 2·13-s − 4·19-s − 25-s + 10·29-s − 8·31-s − 2·37-s + 10·41-s − 12·43-s − 7·49-s + 6·53-s + 8·55-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s − 10·73-s + 8·79-s + 4·83-s + 6·89-s − 8·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s + 0.554·13-s − 0.917·19-s − 1/5·25-s + 1.85·29-s − 1.43·31-s − 0.328·37-s + 1.56·41-s − 1.82·43-s − 49-s + 0.824·53-s + 1.07·55-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(166464\)    =    \(2^{6} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1329.22\)
Root analytic conductor: \(36.4584\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 166464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.541297036\)
\(L(\frac12)\) \(\approx\) \(3.541297036\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.18712639770818, −12.92425644876932, −12.25769574104390, −11.82958748999812, −11.40599583858646, −10.70570417820840, −10.50584881339076, −9.777922682996327, −9.499654882738311, −8.924199375877147, −8.523437296156339, −8.083820977686944, −7.320848353861393, −6.713891593230849, −6.401802245188084, −6.020788006626971, −5.364521634250015, −4.863933013572431, −4.148704637887227, −3.772550158711768, −3.115418219403938, −2.370241001317235, −1.821800739781048, −1.319435745320305, −0.5513820854472790, 0.5513820854472790, 1.319435745320305, 1.821800739781048, 2.370241001317235, 3.115418219403938, 3.772550158711768, 4.148704637887227, 4.863933013572431, 5.364521634250015, 6.020788006626971, 6.401802245188084, 6.713891593230849, 7.320848353861393, 8.083820977686944, 8.523437296156339, 8.924199375877147, 9.499654882738311, 9.777922682996327, 10.50584881339076, 10.70570417820840, 11.40599583858646, 11.82958748999812, 12.25769574104390, 12.92425644876932, 13.18712639770818

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