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Properties

Label 2-2496-39.38-c0-0-0
Degree $2$
Conductor $2496$
Sign $-i$
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  + i·3-s − 1.41·5-s − 1.41i·7-s − 9-s + 13-s − 1.41i·15-s + 1.41i·19-s + 1.41·21-s + 2i·23-s + 1.00·25-s − i·27-s + 1.41i·31-s + 2.00i·35-s + i·39-s + 1.41·41-s + ⋯
L(s)  = 1  + i·3-s − 1.41·5-s − 1.41i·7-s − 9-s + 13-s − 1.41i·15-s + 1.41i·19-s + 1.41·21-s + 2i·23-s + 1.00·25-s − i·27-s + 1.41i·31-s + 2.00i·35-s + i·39-s + 1.41·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7872809066\)
\(L(\frac12)\) \(\approx\) \(0.7872809066\)
\(L(1)\) 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 \( 1 - iT \)
13 \( 1 - T \)
good5 \( 1 + 1.41T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
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

−9.313176348046300302446824794125, −8.449468886174159097299902182177, −7.79112418496234798146560790823, −7.26881381523416255110382907588, −6.14809052640704402813079957376, −5.21506096188547782265622256586, −4.16661449163447086848840944851, −3.79044025715397312760637759074, −3.27366004939959231321470081526, −1.21506878908728488705306832445, 0.60646615724362557697496412137, 2.27023227916355538547473885226, 2.92692902800829250499842427108, 4.06765231584807294370871406566, 5.00523642644847676216737665747, 6.06966967950836823946020506737, 6.54489886318995618017116070996, 7.54009208923847572063007238538, 8.133664610046366956265270406080, 8.761790792952812999029523452585

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