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Properties

Label 1-2925-2925.479-r0-0-0
Degree $1$
Conductor $2925$
Sign $-0.815 + 0.579i$
Analytic cond. $13.5836$
Root an. cond. $13.5836$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.587 − 0.809i)11-s + (−0.913 + 0.406i)14-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + (−0.207 − 0.978i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (−0.207 + 0.978i)28-s + (−0.309 − 0.951i)29-s + (−0.743 + 0.669i)31-s + i·32-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.587 − 0.809i)11-s + (−0.913 + 0.406i)14-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + (−0.207 − 0.978i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (−0.207 + 0.978i)28-s + (−0.309 − 0.951i)29-s + (−0.743 + 0.669i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.815 + 0.579i$
Analytic conductor: \(13.5836\)
Root analytic conductor: \(13.5836\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2925,\ (0:\ ),\ -0.815 + 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4325484300 - 1.355860091i\)
\(L(\frac12)\) \(\approx\) \(-0.4325484300 - 1.355860091i\)
\(L(1)\) \(\approx\) \(0.8107859835 - 0.8510110235i\)
\(L(1)\) \(\approx\) \(0.8107859835 - 0.8510110235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (0.406 - 0.913i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (0.406 + 0.913i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−19.562258900698292371367184183782, −18.639365983144307245147264405018, −18.12648197958274689284084202637, −17.191532913015226920337573823294, −16.56567854543889453656839599566, −16.14410331296968907972345857170, −15.097865154325460439681244299625, −14.82945687695337704785816107733, −14.06117821810019064579549446552, −13.08571224357523283051064906077, −12.645986303619964114730035432030, −12.02570109618735322256262204801, −11.27484163308850216620972421489, −9.9095065929520336611167569108, −9.54809769938615566791927066835, −8.67567866726125776518536187001, −7.8112178289073835852387672633, −7.165369356039702698290541931360, −6.37345259834714687696843973172, −5.7349248158980332476215419564, −5.079244860007432119170644160852, −3.94133459830960554670592856790, −3.52408941022309389281508646533, −2.55670005880831045323561491162, −1.45027021174158468369555497201, 0.38183595058320778777436426387, 1.13900432335126217453428106537, 2.35507707970424419647147179114, 3.13653013435471439363732540203, 3.74332337657306229872449624861, 4.522870966360778050985527829717, 5.444750341463289859658153883833, 6.25857181358559225050290509668, 6.779431432697547161324191598843, 7.884168633618072674146606424889, 8.99247464409298701274074084334, 9.445173716351977431742282726738, 10.30250425909296587716246254953, 10.89455881070720267400830734171, 11.63585611762697131030072249313, 12.35721324937330176919294387053, 13.095685269768379782471315840, 13.59087470713483979789358942597, 14.37412939217439743120203722124, 14.91481516692916955468534864700, 15.961692423183170769605297880788, 16.488542566485700445886820478989, 17.29585685127807475449915050125, 18.265592554469360523384824856450, 18.99568707450862845510697884437

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