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Properties

Label 2-2523-87.23-c0-0-3
Degree $2$
Conductor $2523$
Sign $-0.799 + 0.600i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
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  + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.137 + 0.602i)7-s + (−0.900 − 0.433i)9-s − 12-s + (1.45 − 0.702i)13-s + (−0.900 + 0.433i)16-s + (−0.360 − 1.57i)19-s + (0.556 + 0.268i)21-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + 0.618·28-s + (−0.385 + 0.483i)31-s + (−0.222 + 0.974i)36-s + (−1.45 − 0.702i)37-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.137 + 0.602i)7-s + (−0.900 − 0.433i)9-s − 12-s + (1.45 − 0.702i)13-s + (−0.900 + 0.433i)16-s + (−0.360 − 1.57i)19-s + (0.556 + 0.268i)21-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + 0.618·28-s + (−0.385 + 0.483i)31-s + (−0.222 + 0.974i)36-s + (−1.45 − 0.702i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.799 + 0.600i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062019807\)
\(L(\frac12)\) \(\approx\) \(1.062019807\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.137 - 0.602i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-1.45 + 0.702i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.360 + 1.57i)T + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2} \)
37 \( 1 + (1.45 + 0.702i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.385 + 0.483i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.137 + 0.602i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.45 - 0.702i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.00 - 1.26i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (1.45 + 0.702i)T + (0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2} \)
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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

−8.719269721603570173781310991950, −8.277959037060116206959108451852, −7.03478464279088671034613109677, −6.50583995274447043944455641313, −5.73864635660222897462014068625, −5.17692532281506499739022334287, −3.88298804307348861220733058308, −2.76112606368006930134856483156, −1.85718642364433891097444640928, −0.68919741301323215836102271620, 1.82483046788889447936896001604, 3.30109526937671002171357154317, 3.73670710440203855398306936498, 4.31577758418334292058710297518, 5.38903018215547375727880013399, 6.30707818073237905693459084773, 7.23038715329502711906527708325, 8.153448204026573665598849787855, 8.565394140714248515088762549070, 9.330434680158206463753683081357

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