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Properties

Label 2-2280-2280.1139-c0-0-15
Degree $2$
Conductor $2280$
Sign $1$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + 5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 + 0.707i)10-s − 2i·11-s + (0.707 − 0.707i)12-s − 1.41i·13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + 19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + 5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 + 0.707i)10-s − 2i·11-s + (0.707 − 0.707i)12-s − 1.41i·13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + 19-s + 1.00i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593515113\)
\(L(\frac12)\) \(\approx\) \(1.593515113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + T^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.848175147870554900523024800408, −8.260812883841460062844660125388, −7.52273535330198872359139213415, −6.62682306809598678486736143920, −5.79276965111497701059358053119, −5.69079842260482787959976157862, −4.84556057378320583285549685076, −3.31276704765716271918479716585, −2.70729738919641362871979645479, −1.05860261310693909210248688798, 1.53962113903152399526696618885, 2.33056437466544820814073122667, 3.65280851700053657009761187399, 4.56380653771348253299595284075, 4.98622118720603750882841316633, 5.84689434631890560382277222573, 6.65829110372480489275451063268, 7.23572465696843969893181932198, 9.128634702699688092348506974173, 9.419636437033050020745684956656

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