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Properties

Label 2-2496-39.17-c0-0-0
Degree $2$
Conductor $2496$
Sign $0.0128 + 0.999i$
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  + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)13-s − 1.73i·21-s − 25-s + 0.999·27-s − 1.73i·31-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (1.49 + 0.866i)63-s + (−1.5 − 0.866i)67-s − 1.73i·73-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)13-s − 1.73i·21-s − 25-s + 0.999·27-s − 1.73i·31-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (1.49 + 0.866i)63-s + (−1.5 − 0.866i)67-s − 1.73i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2599444811\)
\(L(\frac12)\) \(\approx\) \(0.2599444811\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.226907870225965268606692651787, −8.310139037466467677814416706263, −7.31891530374740415112900112256, −6.26572621580422915419745239053, −5.86219384488449724756371386568, −5.12471306384362821534328975654, −4.01349331042821314068527253241, −3.25580926179485353574357701053, −2.41786681043685690595772221552, −0.18352339984878109766920219971, 1.35663117935122772900771278979, 2.60746450843408481867753533320, 3.57851722762870678093761302296, 4.56262638672488618336609717268, 5.58668175159702726809228341093, 6.48694838097050973989544920041, 6.86461293021420484100610059941, 7.50891666610498350718347974265, 8.440412931014614590213359938200, 9.407982734760728429576335552575

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