Properties

Degree $2$
Conductor $35490$
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 + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 14-s + 15-s + 16-s − 6·17-s − 18-s + 4·19-s + 20-s − 21-s − 24-s + 25-s + 27-s − 28-s − 6·29-s − 30-s + 4·31-s − 32-s + 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.267·14-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 − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{35490} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 35490,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.23136638120599, −14.80926655015797, −14.05567424414242, −13.63172274204289, −13.19031209886079, −12.64478574609744, −11.95253499089684, −11.44286700632686, −10.84147082622908, −10.26768010207005, −9.794951021802713, −9.294329714389412, −8.709238748015368, −8.503376143289204, −7.531252309232622, −7.114278348479379, −6.720009643391791, −5.776436956456613, −5.516317705299522, −4.400182103324877, −3.961288316065706, −2.987563783948734, −2.557521691542226, −1.839289640477686, −1.053626251642212, 0, 1.053626251642212, 1.839289640477686, 2.557521691542226, 2.987563783948734, 3.961288316065706, 4.400182103324877, 5.516317705299522, 5.776436956456613, 6.720009643391791, 7.114278348479379, 7.531252309232622, 8.503376143289204, 8.709238748015368, 9.294329714389412, 9.794951021802713, 10.26768010207005, 10.84147082622908, 11.44286700632686, 11.95253499089684, 12.64478574609744, 13.19031209886079, 13.63172274204289, 14.05567424414242, 14.80926655015797, 15.23136638120599

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