Properties

Label 2-1350-1.1-c3-0-5
Degree $2$
Conductor $1350$
Sign $1$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s − 14·7-s − 8·8-s − 3·11-s − 47·13-s + 28·14-s + 16·16-s − 39·17-s + 32·19-s + 6·22-s − 99·23-s + 94·26-s − 56·28-s − 51·29-s + 83·31-s − 32·32-s + 78·34-s − 314·37-s − 64·38-s + 108·41-s − 299·43-s − 12·44-s + 198·46-s + 531·47-s − 147·49-s − 188·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.0822·11-s − 1.00·13-s + 0.534·14-s + 1/4·16-s − 0.556·17-s + 0.386·19-s + 0.0581·22-s − 0.897·23-s + 0.709·26-s − 0.377·28-s − 0.326·29-s + 0.480·31-s − 0.176·32-s + 0.393·34-s − 1.39·37-s − 0.273·38-s + 0.411·41-s − 1.06·43-s − 0.0411·44-s + 0.634·46-s + 1.64·47-s − 3/7·49-s − 0.501·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7631366940\)
\(L(\frac12)\) \(\approx\) \(0.7631366940\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 + 3 T + p^{3} T^{2} \)
13 \( 1 + 47 T + p^{3} T^{2} \)
17 \( 1 + 39 T + p^{3} T^{2} \)
19 \( 1 - 32 T + p^{3} T^{2} \)
23 \( 1 + 99 T + p^{3} T^{2} \)
29 \( 1 + 51 T + p^{3} T^{2} \)
31 \( 1 - 83 T + p^{3} T^{2} \)
37 \( 1 + 314 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 + 299 T + p^{3} T^{2} \)
47 \( 1 - 531 T + p^{3} T^{2} \)
53 \( 1 - 564 T + p^{3} T^{2} \)
59 \( 1 + 12 T + p^{3} T^{2} \)
61 \( 1 - 230 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 + 120 T + p^{3} T^{2} \)
73 \( 1 + 1106 T + p^{3} T^{2} \)
79 \( 1 + 739 T + p^{3} T^{2} \)
83 \( 1 - 1086 T + p^{3} T^{2} \)
89 \( 1 - 120 T + p^{3} T^{2} \)
97 \( 1 - 1642 T + p^{3} T^{2} \)
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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

−9.242600091318647644074089138311, −8.566159084959410809212431691149, −7.56509285133778011812258591471, −6.96185619498313739765028299876, −6.09887789655169037927699316238, −5.15088437590206611910291656701, −3.96132193738706139004392196255, −2.87834104011293302782275561667, −1.94446813425265686612180979521, −0.46415502122991868254312454540, 0.46415502122991868254312454540, 1.94446813425265686612180979521, 2.87834104011293302782275561667, 3.96132193738706139004392196255, 5.15088437590206611910291656701, 6.09887789655169037927699316238, 6.96185619498313739765028299876, 7.56509285133778011812258591471, 8.566159084959410809212431691149, 9.242600091318647644074089138311

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