Properties

Label 2-212415-1.1-c1-0-51
Degree $2$
Conductor $212415$
Sign $-1$
Analytic cond. $1696.14$
Root an. cond. $41.1842$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 3·8-s + 9-s + 10-s − 12-s + 6·13-s − 15-s − 16-s − 18-s − 4·19-s + 20-s + 3·24-s + 25-s − 6·26-s + 27-s + 2·29-s + 30-s − 5·32-s − 36-s + 6·37-s + 4·38-s + 6·39-s − 3·40-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.883·32-s − 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.960·39-s − 0.474·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212415\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1696.14\)
Root analytic conductor: \(41.1842\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 212415,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.25384151436251, −12.97063775216524, −12.33116820842411, −11.78988158353711, −11.21883755363805, −10.80795076768606, −10.27230985021937, −10.08521282554944, −9.194377936600174, −8.974086159997587, −8.530135316041587, −8.251250858367649, −7.616079005777383, −7.346816619982047, −6.511627479027732, −6.180092878098436, −5.533042438114129, −4.755043251868245, −4.291991808546731, −3.998750635695289, −3.255546484133713, −2.882882534885343, −1.828193571753863, −1.488780999470562, −0.7711296133260378, 0, 0.7711296133260378, 1.488780999470562, 1.828193571753863, 2.882882534885343, 3.255546484133713, 3.998750635695289, 4.291991808546731, 4.755043251868245, 5.533042438114129, 6.180092878098436, 6.511627479027732, 7.346816619982047, 7.616079005777383, 8.251250858367649, 8.530135316041587, 8.974086159997587, 9.194377936600174, 10.08521282554944, 10.27230985021937, 10.80795076768606, 11.21883755363805, 11.78988158353711, 12.33116820842411, 12.97063775216524, 13.25384151436251

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