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Properties

Label 2-2496-104.77-c1-0-30
Degree $2$
Conductor $2496$
Sign $0.532 - 0.846i$
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  + i·3-s + 3.09·5-s + 0.646i·7-s − 9-s + 1.41·11-s + (2.44 + 2.64i)13-s + 3.09i·15-s + 3.58·17-s + 1.11·19-s − 0.646·21-s − 3.46·23-s + 4.58·25-s − i·27-s − 0.913i·29-s − 1.93i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.38·5-s + 0.244i·7-s − 0.333·9-s + 0.426·11-s + (0.679 + 0.733i)13-s + 0.799i·15-s + 0.868·17-s + 0.256·19-s − 0.140·21-s − 0.722·23-s + 0.916·25-s − 0.192i·27-s − 0.169i·29-s − 0.348i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.532 - 0.846i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (2209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.556167851\)
\(L(\frac12)\) \(\approx\) \(2.556167851\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-2.44 - 2.64i)T \)
good5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 - 0.646iT - 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 - 1.11T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 0.913iT - 29T^{2} \)
31 \( 1 + 1.93iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 - 5.95iT - 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 - 3.74iT - 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 + 9.30T + 59T^{2} \)
61 \( 1 + 0.913iT - 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 + 2.44iT - 71T^{2} \)
73 \( 1 - 5.65iT - 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 8.78iT - 89T^{2} \)
97 \( 1 + 7.89iT - 97T^{2} \)
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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

−9.334683474069541935942025870207, −8.464432724355293856944834278082, −7.58128891567439956941954886535, −6.29269971297072976234435386727, −6.09890997105474956728329407306, −5.19507126483204635657925579925, −4.29722101586598828480843410206, −3.32945762899998192981705078298, −2.26996944205890745248460943220, −1.30821403230402996784600395630, 0.964381111754874319135321554340, 1.83063532270804232912146395668, 2.86279232699078608607904724324, 3.83888480967276077770262712648, 5.12614946603704832365374010357, 5.89866225020810202887914136931, 6.24715924112770650671111852714, 7.29609690249995731734203307896, 7.943850666896921111496199501830, 8.922249663671908249080148767216

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