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Properties

Label 2-2520-840.419-c0-0-1
Degree $2$
Conductor $2520$
Sign $-0.169 - 0.985i$
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 − 5-s + (−0.707 − 0.707i)7-s − i·8-s − i·10-s − 1.41i·11-s + 1.41i·13-s + (0.707 − 0.707i)14-s + 16-s + 2i·19-s + 20-s + 1.41·22-s + 25-s − 1.41·26-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 5-s + (−0.707 − 0.707i)7-s − i·8-s − i·10-s − 1.41i·11-s + 1.41i·13-s + (0.707 − 0.707i)14-s + 16-s + 2i·19-s + 20-s + 1.41·22-s + 25-s − 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7099319359\)
\(L(\frac12)\) \(\approx\) \(0.7099319359\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
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

−9.109244441439507606325730762543, −8.353089806081559064709675463600, −7.74351533637473129979220027752, −7.07264233626288586787618637241, −6.23570960390617009313404909677, −5.73736249159004821913549425264, −4.27914569662652192901389687312, −4.01955183752071016460106636158, −3.12240432197902217176225536400, −1.00726093734955034120714952696, 0.62406343988176230889281986749, 2.40800484068644000838095963685, 2.93575822084934612813616152672, 3.95405021668291684178933797491, 4.77271855840301679466174974236, 5.44527387799530108812710236521, 6.68424244502039142516581207195, 7.52203508868962659469638568746, 8.257499779712492289819397330946, 9.059539254574321504038080088231

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