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Properties

Label 1-2600-2600.2173-r0-0-0
Degree $1$
Conductor $2600$
Sign $-0.109 - 0.994i$
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  + (−0.406 − 0.913i)3-s + (−0.5 − 0.866i)7-s + (−0.669 + 0.743i)9-s + (0.743 − 0.669i)11-s + (−0.406 + 0.913i)17-s + (0.406 − 0.913i)19-s + (−0.587 + 0.809i)21-s + (0.743 − 0.669i)23-s + (0.951 + 0.309i)27-s + (0.913 − 0.406i)29-s + (0.587 + 0.809i)31-s + (−0.913 − 0.406i)33-s + (0.978 + 0.207i)37-s + (−0.207 + 0.978i)41-s + (0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)3-s + (−0.5 − 0.866i)7-s + (−0.669 + 0.743i)9-s + (0.743 − 0.669i)11-s + (−0.406 + 0.913i)17-s + (0.406 − 0.913i)19-s + (−0.587 + 0.809i)21-s + (0.743 − 0.669i)23-s + (0.951 + 0.309i)27-s + (0.913 − 0.406i)29-s + (0.587 + 0.809i)31-s + (−0.913 − 0.406i)33-s + (0.978 + 0.207i)37-s + (−0.207 + 0.978i)41-s + (0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.109 - 0.994i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (2173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ -0.109 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9396367980 - 1.048671258i\)
\(L(\frac12)\) \(\approx\) \(0.9396367980 - 1.048671258i\)
\(L(1)\) \(\approx\) \(0.8764289124 - 0.4111707196i\)
\(L(1)\) \(\approx\) \(0.8764289124 - 0.4111707196i\)

Euler product

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

−19.520338925988677635837256904865, −18.95094952296838927872575387061, −17.817675402777896058385713461353, −17.61732153094601911119173212403, −16.46996539651966934737623438532, −16.11109521597882298077251292963, −15.34911922769344277186407722549, −14.75557596895468423318797671967, −14.06419768471851688364592136094, −12.99739542054950812597317083829, −12.18591910647333840402253989973, −11.68236275590519393151365186265, −11.006607711444494246983318770955, −9.90242907422315269095399327438, −9.53866113329727389546564731865, −8.95271167193517418123158993388, −7.97941321131076393685086113251, −6.85619466415453780915220965423, −6.237965681935028272811271236354, −5.39783817401523763100378927864, −4.75388585019948243872909800883, −3.86520741447734294275895412584, −3.08729231721495333415830815706, −2.21475350937057033445758551975, −0.88206682462381330074679145563, 0.68574644084613302284608131163, 1.227527789271894342629721545422, 2.48093315736793549659485464937, 3.26386327415719330056430431451, 4.2821105060912976432769484063, 5.083455444423830610444356061331, 6.35254908012567339976464418207, 6.45268693271320424678159814172, 7.29760041305411947714290157547, 8.20434444132964083784664499085, 8.82712675454068734680301757141, 9.83857309531320966734191324178, 10.7471829528504373682357336467, 11.22754621892166418216094493934, 12.04651215024881004866279028253, 12.84356378222297601308516785619, 13.410594488629515793548534412605, 13.98577139801902822103343137428, 14.750786649199702899432212260081, 15.82280923423736181612922175846, 16.54170479668162886074449713241, 17.15938965456291954427393626794, 17.62338563615814447255943049111, 18.50393146551035610006639081718, 19.365255051358864044348414464769

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