↑

Properties

Label 2-595-7.4-c1-0-4
Degree $2$
Conductor $595$
Sign $0.790 - 0.612i$
Analytic cond. $4.75109$
Root an. cond. $2.17970$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.783 − 1.35i)2-s + (−0.0392 + 0.0679i)3-s + (−0.227 + 0.394i)4-s + (−0.5 − 0.866i)5-s + 0.122·6-s + (−2.59 + 0.521i)7-s − 2.42·8-s + (1.49 + 2.59i)9-s + (−0.783 + 1.35i)10-s + (−1.25 + 2.17i)11-s + (−0.0178 − 0.0309i)12-s + 0.322·13-s + (2.74 + 3.11i)14-s + 0.0784·15-s + (2.35 + 4.07i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.554 − 0.959i)2-s + (−0.0226 + 0.0392i)3-s + (−0.113 + 0.197i)4-s + (−0.223 − 0.387i)5-s + 0.0501·6-s + (−0.980 + 0.197i)7-s − 0.855·8-s + (0.498 + 0.864i)9-s + (−0.247 + 0.429i)10-s + (−0.379 + 0.656i)11-s + (−0.00515 − 0.00893i)12-s + 0.0895·13-s + (0.732 + 0.831i)14-s + 0.0202·15-s + (0.587 + 1.01i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(4.75109\)
Root analytic conductor: \(2.17970\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{595} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 595,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.519599 + 0.177612i\)
\(L(\frac12)\) \(\approx\) \(0.519599 + 0.177612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.59 - 0.521i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.783 + 1.35i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.0392 - 0.0679i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.25 - 2.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.322T + 13T^{2} \)
19 \( 1 + (-2.51 - 4.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.331 + 0.575i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.620T + 29T^{2} \)
31 \( 1 + (0.101 - 0.175i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.15 - 2.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.93T + 41T^{2} \)
43 \( 1 + 3.56T + 43T^{2} \)
47 \( 1 + (-5.06 - 8.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.425 + 0.737i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.97 - 3.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.05T + 71T^{2} \)
73 \( 1 + (-8.18 + 14.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.02 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.98T + 83T^{2} \)
89 \( 1 + (2.13 + 3.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48388140586009826587400734119, −10.07476627123836452639127553863, −9.323883400568794333212156445637, −8.357392662090383954061992625140, −7.37910731940166975927529410049, −6.21732271989716017081562514095, −5.17750482018148926143813595091, −3.87345506367228163745320642196, −2.69194123716654138969419273049, −1.51492122274317302381609813751, 0.36986383879607468449755296263, 2.92535926808707453509710733517, 3.72361754559613555470543618162, 5.40897416717672289337253952809, 6.49913770372605704025531602307, 6.89675319733506483091089197161, 7.76984803957525880753457318622, 8.829391286736203382249126864350, 9.470495679333248070121910763330, 10.33100310006799990359369327516

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