Properties

Label 2-327990-1.1-c1-0-28
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 13-s − 2·14-s + 15-s + 16-s + 2·17-s − 18-s + 6·19-s + 20-s + 2·21-s − 4·22-s + 7·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.436·21-s − 0.852·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-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: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.577507422\)
\(L(\frac12)\) \(\approx\) \(5.577507422\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.51981704059968, −12.05394589832865, −11.58399631483044, −11.22304928822633, −10.83703615621195, −10.17152154823974, −9.699180552552829, −9.443045014107151, −8.925033896455203, −8.587281810859754, −8.095713688867406, −7.438296232494776, −7.299159820518013, −6.637695143586689, −6.200366217206054, −5.492017904180804, −5.217629575190710, −4.395452754276443, −4.035097269604189, −3.223863357745706, −2.929057553531849, −2.257514669944705, −1.542767817586914, −1.140962824834098, −0.7751514802088237, 0.7751514802088237, 1.140962824834098, 1.542767817586914, 2.257514669944705, 2.929057553531849, 3.223863357745706, 4.035097269604189, 4.395452754276443, 5.217629575190710, 5.492017904180804, 6.200366217206054, 6.637695143586689, 7.299159820518013, 7.438296232494776, 8.095713688867406, 8.587281810859754, 8.925033896455203, 9.443045014107151, 9.699180552552829, 10.17152154823974, 10.83703615621195, 11.22304928822633, 11.58399631483044, 12.05394589832865, 12.51981704059968

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