Properties

Label 2-7800-1.1-c1-0-88
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.82·7-s + 9-s + 0.393·11-s − 13-s + 6.21·17-s − 7.34·19-s − 2.82·21-s + 1.44·23-s + 27-s − 0.828·29-s + 1.65·31-s + 0.393·33-s − 6.78·37-s − 39-s + 3.77·41-s + 2.51·43-s − 3.00·47-s + 1.00·49-s + 6.21·51-s + 9.55·53-s − 7.34·57-s + 0.393·59-s + 11.0·61-s − 2.82·63-s − 15.5·67-s + 1.44·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.06·7-s + 0.333·9-s + 0.118·11-s − 0.277·13-s + 1.50·17-s − 1.68·19-s − 0.617·21-s + 0.301·23-s + 0.192·27-s − 0.153·29-s + 0.297·31-s + 0.0684·33-s − 1.11·37-s − 0.160·39-s + 0.590·41-s + 0.383·43-s − 0.438·47-s + 0.142·49-s + 0.869·51-s + 1.31·53-s − 0.972·57-s + 0.0511·59-s + 1.41·61-s − 0.356·63-s − 1.90·67-s + 0.173·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 0.393T + 11T^{2} \)
17 \( 1 - 6.21T + 17T^{2} \)
19 \( 1 + 7.34T + 19T^{2} \)
23 \( 1 - 1.44T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 + 6.78T + 37T^{2} \)
41 \( 1 - 3.77T + 41T^{2} \)
43 \( 1 - 2.51T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 - 9.55T + 53T^{2} \)
59 \( 1 - 0.393T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 6.56T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + 0.0418T + 97T^{2} \)
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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

−7.38983932414227544302985332787, −6.96305565037860625337245143642, −6.12820333585418415598161679195, −5.55511650876490026472047832885, −4.50966629018661867398739041747, −3.80850160709359404240212361315, −3.10205004050751466307052271770, −2.41349048781533100077599982912, −1.32162337114402594317905037787, 0, 1.32162337114402594317905037787, 2.41349048781533100077599982912, 3.10205004050751466307052271770, 3.80850160709359404240212361315, 4.50966629018661867398739041747, 5.55511650876490026472047832885, 6.12820333585418415598161679195, 6.96305565037860625337245143642, 7.38983932414227544302985332787

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