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Properties

Label 2-1040-260.259-c0-0-0
Degree $2$
Conductor $1040$
Sign $1$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 5-s + 1.00·9-s − 1.41·11-s + 13-s + 1.41·15-s + 1.41·19-s + 1.41·23-s + 25-s − 1.41·31-s + 2.00·33-s − 1.41·39-s + 1.41·43-s − 1.00·45-s + 49-s + 1.41·55-s − 2.00·57-s + 1.41·59-s − 65-s − 2.00·69-s + 1.41·71-s − 1.41·75-s − 0.999·81-s + 2.00·93-s − 1.41·95-s − 2·97-s − 1.41·99-s + ⋯
L(s)  = 1  − 1.41·3-s − 5-s + 1.00·9-s − 1.41·11-s + 13-s + 1.41·15-s + 1.41·19-s + 1.41·23-s + 25-s − 1.41·31-s + 2.00·33-s − 1.41·39-s + 1.41·43-s − 1.00·45-s + 49-s + 1.41·55-s − 2.00·57-s + 1.41·59-s − 65-s − 2.00·69-s + 1.41·71-s − 1.41·75-s − 0.999·81-s + 2.00·93-s − 1.41·95-s − 2·97-s − 1.41·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (1039, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5022058407\)
\(L(\frac12)\) \(\approx\) \(0.5022058407\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−10.58493227577825927102303688820, −9.361512895914561392383342283091, −8.346499680284925639714229568304, −7.45510631718576331858413115010, −6.83760720679743376375199993638, −5.52671865457510743582871966206, −5.26404256201649759613210877238, −4.06742703188410755460416886529, −2.95880759728243062463919871100, −0.877628493329167792748409431113, 0.877628493329167792748409431113, 2.95880759728243062463919871100, 4.06742703188410755460416886529, 5.26404256201649759613210877238, 5.52671865457510743582871966206, 6.83760720679743376375199993638, 7.45510631718576331858413115010, 8.346499680284925639714229568304, 9.361512895914561392383342283091, 10.58493227577825927102303688820

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