Properties

Label 2-327990-1.1-c1-0-36
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 + 8-s + 9-s + 10-s + 4·11-s − 12-s − 13-s − 15-s + 16-s + 4·17-s + 18-s + 20-s + 4·22-s + 2·23-s − 24-s + 25-s − 26-s − 27-s − 30-s − 4·31-s + 32-s − 4·33-s + 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.223·20-s + 0.852·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.182·30-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + 0.685·34-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.75888898000669, −12.46329954942398, −11.81004662495377, −11.69943259963263, −11.10165994060964, −10.68635196452835, −10.06911423097643, −9.821403829588569, −9.212569251494586, −8.820545610554618, −8.184785045385304, −7.500372092774656, −7.202698815734233, −6.674457212851555, −6.111277741493351, −5.920730166524625, −5.286432632896095, −4.763976933224021, −4.483085311101459, −3.580366838312455, −3.450380296970163, −2.730887286772207, −1.926081293801691, −1.496139771347858, −0.9569030472528481, 0, 0.9569030472528481, 1.496139771347858, 1.926081293801691, 2.730887286772207, 3.450380296970163, 3.580366838312455, 4.483085311101459, 4.763976933224021, 5.286432632896095, 5.920730166524625, 6.111277741493351, 6.674457212851555, 7.202698815734233, 7.500372092774656, 8.184785045385304, 8.820545610554618, 9.212569251494586, 9.821403829588569, 10.06911423097643, 10.68635196452835, 11.10165994060964, 11.69943259963263, 11.81004662495377, 12.46329954942398, 12.75888898000669

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