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Properties

Label 2-310464-1.1-c1-0-28
Degree $2$
Conductor $310464$
Sign $1$
Analytic cond. $2479.06$
Root an. cond. $49.7902$
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 + 11-s − 6·13-s − 2·17-s − 8·19-s + 4·23-s − 25-s + 2·29-s + 8·31-s − 6·37-s − 2·41-s − 8·43-s − 4·47-s + 2·53-s − 2·55-s + 12·59-s + 10·61-s + 12·65-s + 12·67-s + 12·71-s − 10·73-s − 8·79-s − 12·83-s + 4·85-s + 10·89-s + 16·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 1.83·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.312·41-s − 1.21·43-s − 0.583·47-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.28·61-s + 1.48·65-s + 1.46·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s − 1.31·83-s + 0.433·85-s + 1.05·89-s + 1.64·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2479.06\)
Root analytic conductor: \(49.7902\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 310464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6387512210\)
\(L(\frac12)\) \(\approx\) \(0.6387512210\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.61499853809400, −12.11769200492812, −11.74266089975753, −11.45832078435328, −10.84618988149637, −10.35877065517698, −9.877994129850838, −9.635015863524433, −8.728242819460748, −8.475898327518072, −8.213775121041411, −7.481212610046694, −7.006830854995141, −6.696965151088800, −6.297047453268597, −5.394980240168084, −4.988960606459846, −4.525120873600140, −4.102323953329624, −3.552125725201456, −2.917142197531854, −2.284697759632389, −1.961495143655530, −0.9497607783380719, −0.2417033310922690, 0.2417033310922690, 0.9497607783380719, 1.961495143655530, 2.284697759632389, 2.917142197531854, 3.552125725201456, 4.102323953329624, 4.525120873600140, 4.988960606459846, 5.394980240168084, 6.297047453268597, 6.696965151088800, 7.006830854995141, 7.481212610046694, 8.213775121041411, 8.475898327518072, 8.728242819460748, 9.635015863524433, 9.877994129850838, 10.35877065517698, 10.84618988149637, 11.45832078435328, 11.74266089975753, 12.11769200492812, 12.61499853809400

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