Properties

Label 2-7800-1.1-c1-0-37
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.57·7-s + 9-s + 5.26·11-s − 13-s − 7.06·17-s + 5.76·19-s − 3.57·21-s − 2.30·23-s − 27-s − 8.14·29-s + 5.87·31-s − 5.26·33-s − 3.38·37-s + 39-s + 4.77·41-s + 7.44·43-s + 9.03·47-s + 5.76·49-s + 7.06·51-s + 5.45·53-s − 5.76·57-s − 8.56·59-s + 8.07·61-s + 3.57·63-s + 8.95·67-s + 2.30·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.35·7-s + 0.333·9-s + 1.58·11-s − 0.277·13-s − 1.71·17-s + 1.32·19-s − 0.779·21-s − 0.480·23-s − 0.192·27-s − 1.51·29-s + 1.05·31-s − 0.917·33-s − 0.555·37-s + 0.160·39-s + 0.745·41-s + 1.13·43-s + 1.31·47-s + 0.823·49-s + 0.989·51-s + 0.749·53-s − 0.763·57-s − 1.11·59-s + 1.03·61-s + 0.450·63-s + 1.09·67-s + 0.277·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.217409871\)
\(L(\frac12)\) \(\approx\) \(2.217409871\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.57T + 7T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
17 \( 1 + 7.06T + 17T^{2} \)
19 \( 1 - 5.76T + 19T^{2} \)
23 \( 1 + 2.30T + 23T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 - 5.87T + 31T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 - 4.77T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 - 9.03T + 47T^{2} \)
53 \( 1 - 5.45T + 53T^{2} \)
59 \( 1 + 8.56T + 59T^{2} \)
61 \( 1 - 8.07T + 61T^{2} \)
67 \( 1 - 8.95T + 67T^{2} \)
71 \( 1 - 9.76T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 1.38T + 79T^{2} \)
83 \( 1 + 9.57T + 83T^{2} \)
89 \( 1 + 4.53T + 89T^{2} \)
97 \( 1 + 7.14T + 97T^{2} \)
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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

−7.77976971788839988200581598402, −7.09436631188799060933279461853, −6.54673371162658005691319258529, −5.67750434039793664449589351225, −5.09969265022228598563053428859, −4.21370361099743871366204623603, −3.92983209274810142688853627879, −2.45875949162226048348238878904, −1.66696896130371117897900726874, −0.813636955478855706489784502271, 0.813636955478855706489784502271, 1.66696896130371117897900726874, 2.45875949162226048348238878904, 3.92983209274810142688853627879, 4.21370361099743871366204623603, 5.09969265022228598563053428859, 5.67750434039793664449589351225, 6.54673371162658005691319258529, 7.09436631188799060933279461853, 7.77976971788839988200581598402

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