Properties

Label 2-327990-1.1-c1-0-25
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 + 7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 2·19-s − 20-s + 21-s + 2·22-s + 3·23-s + 24-s + 25-s + 26-s + 27-s + 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 + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-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

\(L(1)\) \(\approx\) \(8.086039370\)
\(L(\frac12)\) \(\approx\) \(8.086039370\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.63397935572917, −12.18490542426713, −11.78551827499263, −11.24342331722858, −10.92851597745078, −10.51918639932818, −9.846505951754306, −9.210382794628510, −9.052331108005281, −8.431835600256164, −7.871990763085844, −7.592179509858635, −6.977595079376814, −6.558576087818673, −6.149809767351052, −5.342724390704455, −5.058943200332452, −4.395706144017710, −3.954915430144945, −3.627627063877335, −2.988341704275415, −2.348505276236219, −2.012768539611614, −1.067284445760304, −0.7197690430725214, 0.7197690430725214, 1.067284445760304, 2.012768539611614, 2.348505276236219, 2.988341704275415, 3.627627063877335, 3.954915430144945, 4.395706144017710, 5.058943200332452, 5.342724390704455, 6.149809767351052, 6.558576087818673, 6.977595079376814, 7.592179509858635, 7.871990763085844, 8.431835600256164, 9.052331108005281, 9.210382794628510, 9.846505951754306, 10.51918639932818, 10.92851597745078, 11.24342331722858, 11.78551827499263, 12.18490542426713, 12.63397935572917

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