Properties

Label 2-327990-1.1-c1-0-27
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 − 12-s − 13-s + 15-s + 16-s + 6·17-s − 18-s − 4·19-s − 20-s + 24-s + 25-s + 26-s − 27-s − 30-s + 8·31-s − 32-s − 6·34-s + 36-s + 6·37-s + 4·38-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 − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.182·30-s + 1.43·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−12.68139822398230, −12.27738200663639, −11.82624696361269, −11.44486816770922, −11.07281928956392, −10.44638720828478, −10.14960625110320, −9.712830461977108, −9.288189344906855, −8.641727547596088, −8.109805329327297, −7.851056043670875, −7.417675495623947, −6.788734845163652, −6.192683027306882, −6.138788042410980, −5.299744812921643, −4.751993340674888, −4.452336734290186, −3.554948359614089, −3.283586643211854, −2.506256238509403, −1.969010478350989, −1.161633041975332, −0.7478836850769748, 0, 0.7478836850769748, 1.161633041975332, 1.969010478350989, 2.506256238509403, 3.283586643211854, 3.554948359614089, 4.452336734290186, 4.751993340674888, 5.299744812921643, 6.138788042410980, 6.192683027306882, 6.788734845163652, 7.417675495623947, 7.851056043670875, 8.109805329327297, 8.641727547596088, 9.288189344906855, 9.712830461977108, 10.14960625110320, 10.44638720828478, 11.07281928956392, 11.44486816770922, 11.82624696361269, 12.27738200663639, 12.68139822398230

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