Properties

Label 2-327990-1.1-c1-0-26
Degree $2$
Conductor $327990$
Sign $-1$
Analytic cond. $2619.01$
Root an. cond. $51.1762$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s − 12-s + 13-s − 4·14-s + 15-s + 16-s + 18-s + 4·19-s − 20-s + 4·21-s + 6·23-s − 24-s + 25-s + 26-s − 27-s − 4·28-s + 30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s + 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(327990\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2619.01\)
Root analytic conductor: \(51.1762\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{327990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 - T \)
29 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−12.86324879531650, −12.41437297793904, −11.96584976358014, −11.59747316658868, −11.12787469909630, −10.62353019308281, −10.17734588384539, −9.752367764062756, −9.256471487051071, −8.773776464953231, −8.171214939834632, −7.504558627428252, −7.155343785714941, −6.630577270179831, −6.356590285862616, −5.838185051401290, −5.241429412172958, −4.854902463496432, −4.334934372546336, −3.512447467738340, −3.388502885213975, −2.910591122586131, −2.173838724733311, −1.313985738264565, −0.7442093280295308, 0, 0.7442093280295308, 1.313985738264565, 2.173838724733311, 2.910591122586131, 3.388502885213975, 3.512447467738340, 4.334934372546336, 4.854902463496432, 5.241429412172958, 5.838185051401290, 6.356590285862616, 6.630577270179831, 7.155343785714941, 7.504558627428252, 8.171214939834632, 8.773776464953231, 9.256471487051071, 9.752367764062756, 10.17734588384539, 10.62353019308281, 11.12787469909630, 11.59747316658868, 11.96584976358014, 12.41437297793904, 12.86324879531650

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