Properties

Degree $2$
Conductor $58870$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s − 4·11-s − 6·13-s + 14-s + 16-s − 2·17-s + 3·18-s − 20-s + 4·22-s + 25-s + 6·26-s − 28-s − 8·31-s − 32-s + 2·34-s + 35-s − 3·36-s + 10·37-s + 40-s − 2·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58870\)    =    \(2 \cdot 5 \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{58870} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58870,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.72527885456749, −14.33828460037114, −13.33721077509525, −13.12021473768286, −12.48030742725488, −11.95830941688678, −11.49426442993914, −10.95841578279550, −10.54785719736099, −9.786293210852000, −9.620401319323627, −8.828646387031657, −8.426933222350932, −7.760805350338122, −7.445521207037271, −6.939599813441571, −6.123625868767619, −5.682894012910577, −4.925640873804950, −4.579444484692473, −3.481746507796177, −2.970785018446178, −2.441649481142219, −1.846405371604908, −0.5076631430696542, 0, 0.5076631430696542, 1.846405371604908, 2.441649481142219, 2.970785018446178, 3.481746507796177, 4.579444484692473, 4.925640873804950, 5.682894012910577, 6.123625868767619, 6.939599813441571, 7.445521207037271, 7.760805350338122, 8.426933222350932, 8.828646387031657, 9.620401319323627, 9.786293210852000, 10.54785719736099, 10.95841578279550, 11.49426442993914, 11.95830941688678, 12.48030742725488, 13.12021473768286, 13.33721077509525, 14.33828460037114, 14.72527885456749

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