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Properties

Label 1-5600-5600.5221-r1-0-0
Degree $1$
Conductor $5600$
Sign $-0.331 - 0.943i$
Analytic cond. $601.803$
Root an. cond. $601.803$
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  + (0.891 + 0.453i)3-s + (0.587 + 0.809i)9-s + (0.156 + 0.987i)11-s + (0.156 − 0.987i)13-s + (0.309 − 0.951i)17-s + (0.453 + 0.891i)19-s + (−0.587 + 0.809i)23-s + (0.156 + 0.987i)27-s + (−0.891 − 0.453i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)39-s + (−0.587 − 0.809i)41-s + (−0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)3-s + (0.587 + 0.809i)9-s + (0.156 + 0.987i)11-s + (0.156 − 0.987i)13-s + (0.309 − 0.951i)17-s + (0.453 + 0.891i)19-s + (−0.587 + 0.809i)23-s + (0.156 + 0.987i)27-s + (−0.891 − 0.453i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)39-s + (−0.587 − 0.809i)41-s + (−0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(601.803\)
Root analytic conductor: \(601.803\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (5221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (1:\ ),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4213598723 - 0.5945576370i\)
\(L(\frac12)\) \(\approx\) \(0.4213598723 - 0.5945576370i\)
\(L(1)\) \(\approx\) \(1.262298292 + 0.2409632953i\)
\(L(1)\) \(\approx\) \(1.262298292 + 0.2409632953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.891 + 0.453i)T \)
11 \( 1 + (0.156 + 0.987i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.453 - 0.891i)T \)
59 \( 1 + (0.987 + 0.156i)T \)
61 \( 1 + (-0.987 + 0.156i)T \)
67 \( 1 + (0.453 + 0.891i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.453 + 0.891i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.18215497444121113698035687969, −17.093556788765318493013706249436, −16.62121341090356681758354548858, −15.86475819733080284702668822863, −15.06660204246613815185867649784, −14.563630517789064377633756858853, −13.82881734862778400324540240684, −13.42084755411880773214621559374, −12.72179116215276999703719191437, −11.91431830639539628377714787260, −11.34441258508799635940174694100, −10.503761527883901844075907080868, −9.68499404810637239012567957889, −8.9956561665705311020080840870, −8.50911083734730518606701320439, −7.86514951874551381567097597117, −7.02551215171648010967134377110, −6.44568990089495104384002853226, −5.75384736691656526852630010336, −4.69886809805620628308621710531, −3.82849818802741949545805699428, −3.38101448200742633690313861877, −2.43929124230709232734226269061, −1.72981076278364957482893955871, −0.96487093568396006994702997424, 0.0807932819697895448233619431, 1.4071282388868649073433460322, 1.99549012185330021559282250570, 2.95654966898472524483619330041, 3.57173782845310655900778369896, 4.18999522281408015001548296016, 5.268285595234618179314275929658, 5.49779481627684740332091407600, 6.87560643229018093698546282953, 7.45104099795118185816638816151, 7.98295616193310142761478298627, 8.794456373470373724496660574403, 9.47967064103051469557716957264, 10.09136700041013858514308344983, 10.4850836308833012507104722748, 11.55718070541808531555691730115, 12.23872697790067366309841543746, 12.89687241436469469158815532711, 13.716768414965723723573667812177, 14.13814046257918021233654333359, 14.93929122479512428066942389832, 15.41794259664932273486540876359, 16.02462563723554872128160902836, 16.66452111853740223381779943100, 17.6171427206667580356559441870

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