Properties

Label 2-327990-1.1-c1-0-17
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 + 4·7-s + 8-s + 9-s − 10-s − 12-s − 13-s + 4·14-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s − 20-s − 4·21-s + 8·23-s − 24-s + 25-s − 26-s − 27-s + 4·28-s + 30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 0.182·30-s + 1.43·31-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.868009425\)
\(L(\frac12)\) \(\approx\) \(4.868009425\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−12.64603679781798, −11.95106006586379, −11.69026094466167, −11.49572869682921, −10.85013863614599, −10.51002633745729, −10.18164193337369, −9.429957683932685, −8.761115990070720, −8.400521103756670, −7.958634480432956, −7.404633631033155, −7.030085396305424, −6.516577466122280, −5.999207819556739, −5.300605484897704, −5.030670342375546, −4.653269741213946, −4.172480468160605, −3.632281347912578, −2.910582897355153, −2.400708971525416, −1.713541887752158, −1.146580219894488, −0.5743976173162840, 0.5743976173162840, 1.146580219894488, 1.713541887752158, 2.400708971525416, 2.910582897355153, 3.632281347912578, 4.172480468160605, 4.653269741213946, 5.030670342375546, 5.300605484897704, 5.999207819556739, 6.516577466122280, 7.030085396305424, 7.404633631033155, 7.958634480432956, 8.400521103756670, 8.761115990070720, 9.429957683932685, 10.18164193337369, 10.51002633745729, 10.85013863614599, 11.49572869682921, 11.69026094466167, 11.95106006586379, 12.64603679781798

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