Properties

Label 2-327990-1.1-c1-0-1
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 − 12-s − 13-s + 15-s + 16-s + 2·17-s + 18-s − 20-s − 24-s + 25-s − 26-s − 27-s + 30-s + 32-s + 2·34-s + 36-s − 2·37-s + 39-s − 40-s + 2·41-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 + 0.485·17-s + 0.235·18-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 + 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.328·37-s + 0.160·39-s − 0.158·40-s + 0.312·41-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

\(L(1)\) \(\approx\) \(1.187212366\)
\(L(\frac12)\) \(\approx\) \(1.187212366\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.58177262256941, −12.16340056967373, −11.77660177914991, −11.24506758050988, −11.04616610715769, −10.41361842608131, −9.959173487739451, −9.575233880401703, −8.948252757681001, −8.320699325466052, −7.877395267283930, −7.490894861566703, −6.839445864199105, −6.598095474188069, −5.936107841526150, −5.564823543907252, −4.976732524048556, −4.564645871790351, −4.184999291131684, −3.329709123938231, −3.211363279645550, −2.435949154300504, −1.641533523344610, −1.265777796852525, −0.2601717633393157, 0.2601717633393157, 1.265777796852525, 1.641533523344610, 2.435949154300504, 3.211363279645550, 3.329709123938231, 4.184999291131684, 4.564645871790351, 4.976732524048556, 5.564823543907252, 5.936107841526150, 6.598095474188069, 6.839445864199105, 7.490894861566703, 7.877395267283930, 8.320699325466052, 8.948252757681001, 9.575233880401703, 9.959173487739451, 10.41361842608131, 11.04616610715769, 11.24506758050988, 11.77660177914991, 12.16340056967373, 12.58177262256941

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