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Properties

Label 1-2600-2600.853-r0-0-0
Degree $1$
Conductor $2600$
Sign $0.937 - 0.349i$
Analytic cond. $12.0743$
Root an. cond. $12.0743$
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  + (0.951 − 0.309i)3-s − 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.587 + 0.809i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (−0.587 − 0.809i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s − 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.587 + 0.809i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (−0.587 − 0.809i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ 0.937 - 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.403926340 - 0.4335100002i\)
\(L(\frac12)\) \(\approx\) \(2.403926340 - 0.4335100002i\)
\(L(1)\) \(\approx\) \(1.475054183 - 0.1790824555i\)
\(L(1)\) \(\approx\) \(1.475054183 - 0.1790824555i\)

Euler product

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

−19.43270408688384126236017327124, −18.92078037310116501619015949423, −18.149044219368062057398744155933, −17.191159696638443327755793104055, −16.41943588962734068459182431981, −15.77274863559150961235616698963, −15.23219077580165539830039477550, −14.31332287106515301578964738751, −13.868962369125986235736028166156, −13.058398424064107023669994760829, −12.29833206417814964837408337314, −11.69548744386396684099470332498, −10.359691758344779574940978989527, −9.880253876493389864072177816055, −9.41164115528087350865350521929, −8.55959752187072452576820145783, −7.720239396127333956040058446635, −7.030583098573920084010248712395, −6.27725671650668981822683032770, −5.19333947996680965626548697841, −4.30466446756744532082810164047, −3.59487074399181316614414438842, −2.84524421277002643249878014629, −2.07862958552533917118794647538, −0.89949195930638611645179204795, 0.92345582538669303316509290516, 1.689042840014310917638207763790, 3.00480115302346706726435402000, 3.32336892801646279432811007927, 4.04925602504465166493626141749, 5.37446659838912525778359611268, 6.14737834420769744839750150460, 6.94457554456649121827482994872, 7.595600064858411747969977267746, 8.563435083472622273089348461374, 8.98079825777251635620292502249, 10.07989578592857105444459708626, 10.177631664693541340943832871465, 11.87980673768479526985793692875, 11.98139965671394250283275018182, 13.17480532221893816882313101086, 13.55290774085304838641046420633, 14.3113231857573125298287704522, 14.89841179196641002664029346123, 16.06583689399573392062198236418, 16.11150437423882458759817934703, 17.22600408439358840742948783419, 18.08911479634990479528693374321, 18.88182058533449827994019801983, 19.31825016989964546749431205302

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