Properties

Label 2-7800-1.1-c1-0-75
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 − 0.633·7-s + 9-s − 0.177·11-s + 13-s + 4.48·17-s + 0.633·21-s − 6.48·23-s − 27-s − 3.49·29-s + 0.177·33-s + 1.75·37-s − 39-s + 7.67·41-s − 7.49·43-s − 3.18·47-s − 6.59·49-s − 4.48·51-s − 1.75·53-s + 3.93·59-s + 3.01·61-s − 0.633·63-s − 1.62·67-s + 6.48·69-s − 4.41·71-s − 3.85·73-s + 0.112·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.239·7-s + 0.333·9-s − 0.0536·11-s + 0.277·13-s + 1.08·17-s + 0.138·21-s − 1.35·23-s − 0.192·27-s − 0.649·29-s + 0.0309·33-s + 0.288·37-s − 0.160·39-s + 1.19·41-s − 1.14·43-s − 0.465·47-s − 0.942·49-s − 0.628·51-s − 0.241·53-s + 0.512·59-s + 0.385·61-s − 0.0798·63-s − 0.198·67-s + 0.781·69-s − 0.523·71-s − 0.451·73-s + 0.0128·77-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 \( 1 + 0.633T + 7T^{2} \)
11 \( 1 + 0.177T + 11T^{2} \)
17 \( 1 - 4.48T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6.48T + 23T^{2} \)
29 \( 1 + 3.49T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 - 7.67T + 41T^{2} \)
43 \( 1 + 7.49T + 43T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 + 1.75T + 53T^{2} \)
59 \( 1 - 3.93T + 59T^{2} \)
61 \( 1 - 3.01T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 + 3.85T + 73T^{2} \)
79 \( 1 - 9.63T + 79T^{2} \)
83 \( 1 + 1.72T + 83T^{2} \)
89 \( 1 + 0.589T + 89T^{2} \)
97 \( 1 + 4.45T + 97T^{2} \)
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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

−7.60606644400224405371234981745, −6.69191984876913306101670189237, −6.07143934684204828004037915742, −5.53676987629127382346354593870, −4.74806920009578115721255664966, −3.90002217213662972162693252801, −3.24971373905285201124549963162, −2.14132555353948990598335533430, −1.18746793123677191815513396931, 0, 1.18746793123677191815513396931, 2.14132555353948990598335533430, 3.24971373905285201124549963162, 3.90002217213662972162693252801, 4.74806920009578115721255664966, 5.53676987629127382346354593870, 6.07143934684204828004037915742, 6.69191984876913306101670189237, 7.60606644400224405371234981745

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