Properties

Label 2-1380-1.1-c1-0-10
Degree $2$
Conductor $1380$
Sign $-1$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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 − 5-s + 7-s + 9-s − 4.44·11-s + 2.44·13-s + 15-s − 3.44·17-s + 7.34·19-s − 21-s + 23-s + 25-s − 27-s − 9.44·29-s − 1.89·31-s + 4.44·33-s − 35-s − 9.89·37-s − 2.44·39-s − 0.550·41-s − 7.79·43-s − 45-s + 7.34·47-s − 6·49-s + 3.44·51-s + 4.34·53-s + 4.44·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.34·11-s + 0.679·13-s + 0.258·15-s − 0.836·17-s + 1.68·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.75·29-s − 0.341·31-s + 0.774·33-s − 0.169·35-s − 1.62·37-s − 0.392·39-s − 0.0859·41-s − 1.18·43-s − 0.149·45-s + 1.07·47-s − 0.857·49-s + 0.483·51-s + 0.597·53-s + 0.599·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 - T + 7T^{2} \)
11 \( 1 + 4.44T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 - 7.34T + 19T^{2} \)
29 \( 1 + 9.44T + 29T^{2} \)
31 \( 1 + 1.89T + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 + 0.550T + 41T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 - 4.34T + 53T^{2} \)
59 \( 1 + 6.55T + 59T^{2} \)
61 \( 1 - 0.449T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 + 7.34T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 - 4.89T + 97T^{2} \)
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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

−9.161965411368617917876431293385, −8.285837061711125172767237981094, −7.49922815042511903546591410749, −6.86960068734076945920081993824, −5.54676928091779752859014574460, −5.21125257439423591242248807732, −4.05149555290176701628881844469, −3.03726471111768539137008212033, −1.61359930541872423291434609629, 0, 1.61359930541872423291434609629, 3.03726471111768539137008212033, 4.05149555290176701628881844469, 5.21125257439423591242248807732, 5.54676928091779752859014574460, 6.86960068734076945920081993824, 7.49922815042511903546591410749, 8.285837061711125172767237981094, 9.161965411368617917876431293385

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