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Properties

Label 2-2520-280.69-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.260510600\)
\(L(\frac12)\) \(\approx\) \(1.260510600\)
\(L(1)\) 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

−9.042427766819186470553557295708, −8.370683344412948174289370838900, −7.81537087955322464454039688412, −6.82309158409667021732299080566, −5.87576438843376658651949441175, −5.41834087466413072585229649494, −4.93841222341856648093892857258, −3.63773937410631060655005934609, −2.57122152479373679057078250940, −1.04773547050321697309876268420, 1.25481888847999473617822282080, 2.20431174424225376329667952216, 3.24714793227316224933648503366, 3.98230081103615518005565762796, 4.88341680713986146242717727265, 5.78306524062205899355110329751, 6.82763777237242469461493147675, 7.37551162160778305985684259661, 8.431327821139482313600047996792, 9.384924246724247000453699452686

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