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Properties

Label 2-2496-312.83-c0-0-0
Degree $2$
Conductor $2496$
Sign $0.471 - 0.881i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  − 3-s + (−1 + i)7-s + 9-s − i·13-s + (1 − i)19-s + (1 − i)21-s + i·25-s − 27-s + (1 + i)31-s + (−1 + i)37-s + i·39-s + 2i·43-s − i·49-s + (−1 + i)57-s + 2i·61-s + ⋯
L(s)  = 1  − 3-s + (−1 + i)7-s + 9-s − i·13-s + (1 − i)19-s + (1 − i)21-s + i·25-s − 27-s + (1 + i)31-s + (−1 + i)37-s + i·39-s + 2i·43-s − i·49-s + (−1 + i)57-s + 2i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6932529888\)
\(L(\frac12)\) \(\approx\) \(0.6932529888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 + iT \)
good5 \( 1 - iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.406341518427697898461927824375, −8.558885721645276825035866258434, −7.55438943503388609809169084480, −6.75680233350580324166495678319, −6.12702657750630376379904436122, −5.34098408582906122664413196981, −4.83094797818488488928753565992, −3.41348833879036424817153220918, −2.73869407454204586645458863627, −1.13384196265056487221457091608, 0.62333462382742577023659708620, 2.00750035431787600840391285944, 3.58734273369507363531222204989, 4.08998479234489716621711418651, 5.09098650939926648015863496159, 5.99948092411251378989704605809, 6.66345525611340441945464359417, 7.19729776648873898314850890504, 8.036966417482462788692381946811, 9.232710527115376021143863825471

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