Properties

Label 2-2520-280.69-c0-0-0
Degree $2$
Conductor $2520$
Sign $-0.707 + 0.707i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·7-s i·8-s + (−0.707 − 0.707i)10-s + 1.41i·13-s − 14-s + 16-s − 1.41·19-s + (0.707 − 0.707i)20-s − 1.00i·25-s − 1.41·26-s i·28-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.707 + 0.707i)5-s + i·7-s i·8-s + (−0.707 − 0.707i)10-s + 1.41i·13-s − 14-s + 16-s − 1.41·19-s + (0.707 − 0.707i)20-s − 1.00i·25-s − 1.41·26-s i·28-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5398801684\)
\(L(\frac12)\) \(\approx\) \(0.5398801684\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good11 \( 1 - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.179242798482861785236376953194, −8.775658579597500283025777972187, −7.981390640158427937321081169119, −7.23666946185553432540306932641, −6.41409372856590097726043341641, −6.07354163042367560984761999414, −4.80689444032175955156361529305, −4.23040721017499197367520859141, −3.22677706503195506767150351788, −2.01742807936534045086476165387, 0.36638108559426624345244067375, 1.50929048096616552089819148766, 2.90396793095614900778452243259, 3.75818322319115444571216853515, 4.43369372567518972194928568396, 5.13361758867977738076735103171, 6.16345406032983810521421051557, 7.42459472298516438936747355682, 8.035627739889890851360047391194, 8.613559001816800905084181536348

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