Properties

Label 2-327990-1.1-c1-0-2
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 − 3·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 13-s + 3·14-s + 15-s + 16-s − 3·17-s − 18-s − 4·19-s + 20-s − 3·21-s − 4·22-s − 3·23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-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 + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.654·21-s − 0.852·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-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

\(L(1)\) \(\approx\) \(1.030097678\)
\(L(\frac12)\) \(\approx\) \(1.030097678\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.61884171166484, −12.31237129133725, −11.42385363156051, −11.32713417426404, −10.68169694445640, −10.13439168366944, −9.775724241919132, −9.381376072216617, −8.992779435372097, −8.630101731377285, −8.173042841464901, −7.505418015342097, −6.981278241180586, −6.615960172251302, −6.181255404154209, −5.903681661663458, −5.037472451216470, −4.383640792577342, −3.830497871847662, −3.416697502901539, −2.918528277522991, −2.057235179487692, −1.908940175291774, −1.163634048011649, −0.2870095142621159, 0.2870095142621159, 1.163634048011649, 1.908940175291774, 2.057235179487692, 2.918528277522991, 3.416697502901539, 3.830497871847662, 4.383640792577342, 5.037472451216470, 5.903681661663458, 6.181255404154209, 6.615960172251302, 6.981278241180586, 7.505418015342097, 8.173042841464901, 8.630101731377285, 8.992779435372097, 9.381376072216617, 9.775724241919132, 10.13439168366944, 10.68169694445640, 11.32713417426404, 11.42385363156051, 12.31237129133725, 12.61884171166484

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