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Properties

Label 2-2563-2563.2562-c0-0-1
Degree $2$
Conductor $2563$
Sign $-1$
Analytic cond. $1.27910$
Root an. cond. $1.13097$
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  + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 1.41i·5-s − 2.00·6-s − 1.00·9-s + 2.00·10-s + 11-s − 1.41i·12-s + 2.00·15-s − 0.999·16-s + 17-s − 1.41i·18-s + 1.41i·19-s + 1.41i·20-s + ⋯
L(s)  = 1  + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 1.41i·5-s − 2.00·6-s − 1.00·9-s + 2.00·10-s + 11-s − 1.41i·12-s + 2.00·15-s − 0.999·16-s + 17-s − 1.41i·18-s + 1.41i·19-s + 1.41i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2563\)    =    \(11 \cdot 233\)
Sign: $-1$
Analytic conductor: \(1.27910\)
Root analytic conductor: \(1.13097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2563} (2562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2563,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227715481\)
\(L(\frac12)\) \(\approx\) \(1.227715481\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
233 \( 1 + T \)
good2 \( 1 - 1.41iT - T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−9.304686852810309343202514046506, −8.664110152477491435873076411038, −8.070962671068054251564340398007, −7.27052729667944936613773270747, −6.03383198274305542041294048163, −5.62652386568094157768558564436, −4.86128623090652961562820415705, −4.22110170255850010998219815902, −3.51174764219094020858860488339, −1.53775904386848255414056041634, 0.876317126297140593477752072465, 2.02481998271696471342197195870, 2.58030264663678207229424792172, 3.45035911218656949433086139805, 4.28609012853606747101164254145, 5.90243555507561075622722313913, 6.50272788793367664639213024025, 7.20700548727290154519904787508, 7.70124270646371190394941336614, 8.870278931697001333929903178522

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