↑

Properties

Label 1-2925-2925.1094-r0-0-0
Degree $1$
Conductor $2925$
Sign $-0.589 + 0.807i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s − i·7-s + (−0.951 + 0.309i)8-s + (0.406 − 0.913i)11-s + (−0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.669 + 0.743i)17-s + (−0.743 − 0.669i)19-s + (−0.669 − 0.743i)22-s + (0.809 − 0.587i)23-s + (−0.743 + 0.669i)28-s + (0.978 − 0.207i)29-s + (−0.743 − 0.669i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s − i·7-s + (−0.951 + 0.309i)8-s + (0.406 − 0.913i)11-s + (−0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.669 + 0.743i)17-s + (−0.743 − 0.669i)19-s + (−0.669 − 0.743i)22-s + (0.809 − 0.587i)23-s + (−0.743 + 0.669i)28-s + (0.978 − 0.207i)29-s + (−0.743 − 0.669i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.589 + 0.807i$
Analytic conductor: \(13.5836\)
Root analytic conductor: \(13.5836\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2925,\ (0:\ ),\ -0.589 + 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5068571070 - 0.9973940174i\)
\(L(\frac12)\) \(\approx\) \(-0.5068571070 - 0.9973940174i\)
\(L(1)\) \(\approx\) \(0.7026268772 - 0.7856429872i\)
\(L(1)\) \(\approx\) \(0.7026268772 - 0.7856429872i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.4820221224232438957850495455, −18.65760931947904662472217831280, −17.97465135910452694813969450985, −17.51594042224426315693427407314, −16.64780678509678026901680050002, −15.989772171367341140941861084765, −15.2107648420898319978813350710, −14.90324772917695805475826318735, −14.07939986571259661973251412375, −13.30895040084629991249746395175, −12.4743763453210246581774424105, −12.12549103507286212679560416811, −11.24271125651904567773640539724, −10.12516228203328163546864500263, −9.183838895964533810766910804735, −8.883361885592263594135743338109, −7.9514901633561132876397826619, −7.14654901775710862328984534581, −6.49802301507677936775399234895, −5.7831148875991989638696698940, −4.886447095409723787637722195942, −4.4462577742125121140362496510, −3.31528812437699932563989414093, −2.56994987179793341454391343911, −1.48293787243849668376458694490, 0.3145290163863201556807853607, 1.14800913398400674539069776833, 2.16693951501093463066465034989, 3.01669876282890560715127976231, 3.95971531756773223814032321060, 4.317706295629903773695556759758, 5.30515416453529167469350073062, 6.25392525553894853948282646786, 6.81152749480005407940700544780, 7.99333822768121186648651875494, 8.81743146152536589876022793601, 9.36462135898001088193232470221, 10.45374966030961737052052952272, 10.89123600833061044570696650321, 11.31052753627607658704561953364, 12.43943678963595900311443419215, 12.971581094546818953921414988484, 13.647101140367394568997615442371, 14.25532812917383655010149031451, 14.90509002826905717067567285874, 15.78615030427374375466195480166, 16.70851999889937718886854239702, 17.384557247047947120496927515783, 17.96812547276552855447302329536, 19.034853003762297604567572498946

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