Properties

Label 2-2548-364.115-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.482 - 0.875i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.17 − 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s + 1.21·10-s + (−0.130 − 0.991i)13-s + (0.500 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.17 + 0.315i)20-s + (0.417 + 0.241i)25-s + (0.382 + 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.17 − 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s + 1.21·10-s + (−0.130 − 0.991i)13-s + (0.500 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.17 + 0.315i)20-s + (0.417 + 0.241i)25-s + (0.382 + 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.482 - 0.875i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.482 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2409476770\)
\(L(\frac12)\) \(\approx\) \(0.2409476770\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
13 \( 1 + (0.130 + 0.991i)T \)
good3 \( 1 + T^{2} \)
5 \( 1 + (1.17 + 0.315i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (1.91 + 0.513i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 - 1.58iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.53 - 0.410i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \)
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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

−8.965018596184294875409134488718, −8.547267015924319275119492440249, −7.968229026462835981901761098588, −7.31522655822889284942978710336, −6.43084377861565523545734855768, −5.54037040525370852952476693406, −4.78837143492362589180903728589, −3.42775522796061511811109942993, −2.77167420031668173756156906995, −1.18771722730415636478809701578, 0.23659286314471220515139068177, 1.97183649968140835903726258331, 3.00200025523759241832328764188, 3.73255511884505025519744355038, 4.73411889114349590040707239737, 6.05343035827264798286357389692, 6.73392508708023373119945620798, 7.53000420155429102213624010058, 8.174867591214984942794519681052, 8.689214384040009448060020346077

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