Properties

Label 2-7800-1.1-c1-0-90
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 + 2.41·7-s + 9-s + 0.414·11-s − 13-s + 3.82·17-s − 2·19-s − 2.41·21-s − 0.828·23-s − 27-s − 8.65·29-s + 4.41·31-s − 0.414·33-s − 7.65·37-s + 39-s + 5.65·41-s − 10.4·43-s − 2.07·47-s − 1.17·49-s − 3.82·51-s − 5.82·53-s + 2·57-s − 6.41·59-s − 1.82·61-s + 2.41·63-s + 0.757·67-s + 0.828·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.912·7-s + 0.333·9-s + 0.124·11-s − 0.277·13-s + 0.928·17-s − 0.458·19-s − 0.526·21-s − 0.172·23-s − 0.192·27-s − 1.60·29-s + 0.792·31-s − 0.0721·33-s − 1.25·37-s + 0.160·39-s + 0.883·41-s − 1.59·43-s − 0.302·47-s − 0.167·49-s − 0.536·51-s − 0.800·53-s + 0.264·57-s − 0.835·59-s − 0.234·61-s + 0.304·63-s + 0.0925·67-s + 0.0997·69-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 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 - 0.414T + 11T^{2} \)
17 \( 1 - 3.82T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 - 4.41T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 2.07T + 47T^{2} \)
53 \( 1 + 5.82T + 53T^{2} \)
59 \( 1 + 6.41T + 59T^{2} \)
61 \( 1 + 1.82T + 61T^{2} \)
67 \( 1 - 0.757T + 67T^{2} \)
71 \( 1 + 7.65T + 71T^{2} \)
73 \( 1 - 9.31T + 73T^{2} \)
79 \( 1 + 8.82T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−7.65505552484139893910982204867, −6.76148596586763014391602417703, −6.10534376852989647463977389456, −5.29349847439803882073004442338, −4.86700783069264527152267584865, −4.00923235246730581677128086819, −3.19958270232897897499669005635, −2.00770662598978189488872407011, −1.32737307812327504819314845613, 0, 1.32737307812327504819314845613, 2.00770662598978189488872407011, 3.19958270232897897499669005635, 4.00923235246730581677128086819, 4.86700783069264527152267584865, 5.29349847439803882073004442338, 6.10534376852989647463977389456, 6.76148596586763014391602417703, 7.65505552484139893910982204867

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