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Properties

Label 1-2652-2652.887-r1-0-0
Degree $1$
Conductor $2652$
Sign $-0.552 + 0.833i$
Analytic cond. $284.996$
Root an. cond. $284.996$
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  + (0.382 − 0.923i)5-s + (0.608 − 0.793i)7-s + (0.130 − 0.991i)11-s + (0.965 + 0.258i)19-s + (−0.130 + 0.991i)23-s + (−0.707 − 0.707i)25-s + (−0.608 − 0.793i)29-s + (0.923 + 0.382i)31-s + (−0.5 − 0.866i)35-s + (−0.793 + 0.608i)37-s + (−0.991 − 0.130i)41-s + (−0.258 + 0.965i)43-s − i·47-s + (−0.258 − 0.965i)49-s + (−0.707 + 0.707i)53-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)5-s + (0.608 − 0.793i)7-s + (0.130 − 0.991i)11-s + (0.965 + 0.258i)19-s + (−0.130 + 0.991i)23-s + (−0.707 − 0.707i)25-s + (−0.608 − 0.793i)29-s + (0.923 + 0.382i)31-s + (−0.5 − 0.866i)35-s + (−0.793 + 0.608i)37-s + (−0.991 − 0.130i)41-s + (−0.258 + 0.965i)43-s − i·47-s + (−0.258 − 0.965i)49-s + (−0.707 + 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(284.996\)
Root analytic conductor: \(284.996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (887, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (1:\ ),\ -0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06749890898 - 0.1257215021i\)
\(L(\frac12)\) \(\approx\) \(-0.06749890898 - 0.1257215021i\)
\(L(1)\) \(\approx\) \(1.019624478 - 0.3208174236i\)
\(L(1)\) \(\approx\) \(1.019624478 - 0.3208174236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (0.608 - 0.793i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (-0.130 + 0.991i)T \)
29 \( 1 + (-0.608 - 0.793i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.793 + 0.608i)T \)
41 \( 1 + (-0.991 - 0.130i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.258 + 0.965i)T \)
61 \( 1 + (0.608 - 0.793i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.130 - 0.991i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.991 + 0.130i)T \)
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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

−19.538400666400233974168769735016, −18.66727257557683212890490267638, −18.24812890107866126813846536609, −17.65653560845930217957794099576, −16.97840921427742046041222452924, −15.89846784325335273437151804836, −15.22512081585649483407992160729, −14.69810728349784531577491075047, −14.084258093614271759845302315218, −13.28695414215505750790571966357, −12.290490981611573546673162093, −11.79824352674957445357632499063, −10.981638940274411894889259981785, −10.23396084202189667995628229347, −9.572970773523854588867426706486, −8.77713497340459432521054367984, −7.91780239374519707077293171637, −7.06310299307821733241496120984, −6.53966000585706664376794515410, −5.47912912159124081090922671227, −4.978552860608855066834405809552, −3.88250841398502188837128285249, −2.93237821972940931619783790792, −2.17608410588366243346952101095, −1.51747352754382648648584110099, 0.02132765666336723807259592743, 1.18471022775783140802013263547, 1.46758179964894223265686522250, 2.873896216989717586948307504197, 3.76447645199973913666675628127, 4.5693748509983867588752228046, 5.33956359307244831079102709227, 5.97509759509829754134291756908, 6.97824574707686378679981285861, 7.96484886194575321882776157750, 8.312313873147049571946181187340, 9.34352860973675710806568620767, 9.89113997310923563009006078264, 10.80360593732178219806418737852, 11.5795232310524162161181529484, 12.11531155635881201379714502828, 13.222948911578890749503478950981, 13.71400962598537740519011788716, 14.112585963690380406008437380910, 15.21409899211111797010574835889, 16.033759098767343645597348139728, 16.59182376303766583224571881852, 17.33223669602695794695128276037, 17.71234688860321519777155780920, 18.73964294869320440721693961890

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