Properties

Label 2-327990-1.1-c1-0-13
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 − 8-s + 9-s − 10-s − 4·11-s + 12-s + 13-s + 15-s + 16-s + 6·17-s − 18-s − 4·19-s + 20-s + 4·22-s + 8·23-s − 24-s + 25-s − 26-s + 27-s − 30-s + 8·31-s − 32-s − 4·33-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 + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-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 − 0.696·33-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

\(L(1)\) \(\approx\) \(2.911446767\)
\(L(\frac12)\) \(\approx\) \(2.911446767\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.69666161476532, −12.23873510877034, −11.57204498721539, −11.04048701717502, −10.74976132913014, −10.20959072177625, −9.859455728199230, −9.391161982420879, −9.058442255905660, −8.296176310352168, −8.058124681837676, −7.794545429873771, −7.142651974048149, −6.567265856924974, −6.215757851298796, −5.586081181088183, −5.059853683412893, −4.633201427726457, −3.898695857620632, −3.200968690333928, −2.699674072292087, −2.552976055572017, −1.616682428702986, −1.139519150207706, −0.5172552967168294, 0.5172552967168294, 1.139519150207706, 1.616682428702986, 2.552976055572017, 2.699674072292087, 3.200968690333928, 3.898695857620632, 4.633201427726457, 5.059853683412893, 5.586081181088183, 6.215757851298796, 6.567265856924974, 7.142651974048149, 7.794545429873771, 8.058124681837676, 8.296176310352168, 9.058442255905660, 9.391161982420879, 9.859455728199230, 10.20959072177625, 10.74976132913014, 11.04048701717502, 11.57204498721539, 12.23873510877034, 12.69666161476532

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