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Properties

Label 1-2600-2600.1883-r1-0-0
Degree $1$
Conductor $2600$
Sign $0.994 + 0.101i$
Analytic cond. $279.408$
Root an. cond. $279.408$
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  + (−0.743 + 0.669i)3-s + (−0.5 − 0.866i)7-s + (0.104 − 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.743 + 0.669i)17-s + (−0.743 − 0.669i)19-s + (0.951 + 0.309i)21-s + (0.994 − 0.104i)23-s + (0.587 + 0.809i)27-s + (0.669 + 0.743i)29-s + (0.951 − 0.309i)31-s + (0.669 − 0.743i)33-s + (0.913 − 0.406i)37-s + (0.406 + 0.913i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)3-s + (−0.5 − 0.866i)7-s + (0.104 − 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.743 + 0.669i)17-s + (−0.743 − 0.669i)19-s + (0.951 + 0.309i)21-s + (0.994 − 0.104i)23-s + (0.587 + 0.809i)27-s + (0.669 + 0.743i)29-s + (0.951 − 0.309i)31-s + (0.669 − 0.743i)33-s + (0.913 − 0.406i)37-s + (0.406 + 0.913i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.994 + 0.101i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ 0.994 + 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068615108 + 0.05424968561i\)
\(L(\frac12)\) \(\approx\) \(1.068615108 + 0.05424968561i\)
\(L(1)\) \(\approx\) \(0.7324778137 + 0.06240657166i\)
\(L(1)\) \(\approx\) \(0.7324778137 + 0.06240657166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.743 + 0.669i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (0.994 - 0.104i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.406 + 0.913i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.994 - 0.104i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (0.207 + 0.978i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.994 - 0.104i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−19.00247707648195451138947425398, −18.53887431569578994684960358858, −17.90342680044403289143905059996, −17.03622843627457159016876737530, −16.44414607104443669706005136966, −15.68682398462673571461500854200, −15.11424081445184702784216075780, −13.97270003810647776509834671455, −13.37482554597514207391966368145, −12.526311669912746265642826680735, −12.211290140128406417334824651979, −11.36974054864592334392021790717, −10.55848773509528203469453048659, −9.92065001118679548446275542511, −8.944467455093802019547105891691, −8.056789458480217679146612841, −7.50502358730505798120064023047, −6.46856378035715420406506417813, −5.96439268270272804177022535195, −5.21044654598541238334485327845, −4.512364622874810690706413977622, −3.03447545060420293286343360630, −2.558337955267242688873270466460, −1.469415527923891581447699440386, −0.44936977770502493488243510604, 0.42851939392467626237992676849, 1.28982908520892018927517825355, 2.79433977294364378378266350969, 3.42577083795414611993277275236, 4.5061327031900245146488386754, 4.86140683394552017985109785268, 5.96558546681356362313798750976, 6.53700113632961162495724716283, 7.39042502056364779890213303727, 8.25202011358366159495032571464, 9.23286013651171731755353541065, 10.009589104341893306599638186329, 10.55247122190089564717001744734, 11.03655378719585679188762009667, 11.98561056954853470573936188942, 12.9050869111388510102957905543, 13.19946304231748722616107197593, 14.386320488711808795246913338444, 15.06719260015670505984221859804, 15.753654551361037830212266916319, 16.46120202701994055586544766775, 16.99367671703120407716453390767, 17.60643764323858061572797777994, 18.41531127969756044442772825006, 19.20855418579377455513485292910

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