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Properties

Label 1-5160-5160.2243-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.836 + 0.547i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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  + (−0.866 + 0.5i)7-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s − 41-s − i·47-s + (0.5 − 0.866i)49-s + (−0.866 − 0.5i)53-s + 59-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s − 41-s − i·47-s + (0.5 − 0.866i)49-s + (−0.866 − 0.5i)53-s + 59-s + (−0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8275023252 + 0.2468710620i\)
\(L(\frac12)\) \(\approx\) \(0.8275023252 + 0.2468710620i\)
\(L(1)\) \(\approx\) \(0.7800918437 + 0.05005910419i\)
\(L(1)\) \(\approx\) \(0.7800918437 + 0.05005910419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 - iT \)
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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

−17.74892012287266789337658969479, −17.39877864986017102229074660860, −16.449270400909009631059782225158, −16.092730513769728746183182903849, −15.26009062169301551979356407809, −14.72097704148035477482749025229, −13.81585433237390092665835833097, −13.25988662700765857379652463803, −12.547542369240608573893602522863, −12.19957351802254040882966627283, −10.99921399427298327402762066261, −10.51757937999186783823615339860, −9.8735400583062078675989814461, −9.32558717946233572847124901085, −8.29080941536648264837964614336, −7.686450838278322151604241181646, −7.07362256732115219440115574401, −6.27038492132977988018054797175, −5.50136916885689022094085768882, −4.8463897898368110756868029238, −3.94656977934174233634915859395, −3.11655669381312294944034194369, −2.61000880034857600009017567442, −1.52208415489483489821848015227, −0.40126199242332344890043951544, 0.52072904729740570657859962447, 1.99375317783778035619306199651, 2.47215250216870302158194205160, 3.30721714287845242519558014062, 4.121451565218316876861386176567, 4.993005853796574740003030527084, 5.71381828269889070763794910906, 6.31721777689034940745145899125, 7.12873356064622656652558220551, 7.95130370627960046015507717261, 8.39958263088995092491751829344, 9.57303444651046289140926829951, 9.860622676924286254666369262926, 10.42352623709052029601495497093, 11.630817140144075351262884335700, 11.95477923591729406306498113404, 12.799428918295197579605908291709, 13.2307860648816767323762564657, 14.12849553676748352978048543122, 14.76077481775710758887181297962, 15.44962491099894522179697448874, 16.105622956490030767451130621764, 16.688723498702344043079111806143, 17.23010662127112647245747249285, 18.30790397338433510973977939373

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