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Properties

Label 2-2496-13.12-c1-0-44
Degree $2$
Conductor $2496$
Sign $-0.277 + 0.960i$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
Motivic weight $1$
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 − 3.46i·7-s + 9-s + 3.46i·11-s + (1 − 3.46i)13-s − 6·17-s − 3.46i·19-s − 3.46i·21-s + 5·25-s + 27-s − 6·29-s − 3.46i·31-s + 3.46i·33-s − 6.92i·37-s + (1 − 3.46i)39-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.30i·7-s + 0.333·9-s + 1.04i·11-s + (0.277 − 0.960i)13-s − 1.45·17-s − 0.794i·19-s − 0.755i·21-s + 25-s + 0.192·27-s − 1.11·29-s − 0.622i·31-s + 0.603i·33-s − 1.13i·37-s + (0.160 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.688111609\)
\(L(\frac12)\) \(\approx\) \(1.688111609\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−8.760958007374426773943273257450, −7.79939317146945905926604743385, −7.20322215129980292762135946322, −6.71991931490483496745692981924, −5.47869550228222671709826868017, −4.44781288339584667435094545926, −3.99197194374973620356758048563, −2.86897579101799653143808972117, −1.87763433979001016339852324693, −0.50127494790889295092305902654, 1.54519855111347640862829567898, 2.49603684641702680291154491293, 3.31720134533750425500634746113, 4.31937596664120967441103809626, 5.24105676508759016035412698740, 6.18097515676922871963785210228, 6.67979676043420737769096360379, 7.82147529372751813440576293993, 8.685986583575293769474638392246, 8.878723343131379539166875428448

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