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Properties

Label 2-2448-612.475-c0-0-3
Degree $2$
Conductor $2448$
Sign $-0.642 + 0.766i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  + (0.258 − 0.965i)3-s + (0.707 − 1.22i)7-s + (−0.866 − 0.499i)9-s + (0.258 − 0.448i)11-s + (−0.866 − 1.5i)13-s − 17-s + (−0.999 − i)21-s + (0.965 + 1.67i)23-s + (−0.5 + 0.866i)25-s + (−0.707 + 0.707i)27-s + (−0.258 − 0.448i)31-s + (−0.366 − 0.366i)33-s + (−1.67 + 0.448i)39-s + (−0.499 − 0.866i)49-s + (−0.258 + 0.965i)51-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (0.707 − 1.22i)7-s + (−0.866 − 0.499i)9-s + (0.258 − 0.448i)11-s + (−0.866 − 1.5i)13-s − 17-s + (−0.999 − i)21-s + (0.965 + 1.67i)23-s + (−0.5 + 0.866i)25-s + (−0.707 + 0.707i)27-s + (−0.258 − 0.448i)31-s + (−0.366 − 0.366i)33-s + (−1.67 + 0.448i)39-s + (−0.499 − 0.866i)49-s + (−0.258 + 0.965i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.205605666\)
\(L(\frac12)\) \(\approx\) \(1.205605666\)
\(L(1)\) 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 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 + T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.93T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.711152871209079642631190813169, −7.75785604206388102412758765551, −7.51999017993087900914465267508, −6.80331342637536573915831385691, −5.71331316836472226988419583928, −5.04405839141590223722328553086, −3.83631749904397239549551427847, −3.04228161837163890650834386701, −1.84272996912631111620898808891, −0.76952096587192269131377161077, 2.16600079847782842015311866959, 2.49431754367391393524732551280, 4.00901566478394379248886478101, 4.68446057874211093376870424412, 5.16743903059359849783118305626, 6.31772833005243078756692751593, 7.01525331801178029404019562817, 8.229825387098319708080966693731, 8.782195133947568938028687384853, 9.252128656233054720953948072250

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