↑

Properties

Label 2-755-151.118-c1-0-19
Degree $2$
Conductor $755$
Sign $0.513 - 0.858i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
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  + (0.5 − 0.866i)2-s + 0.638·3-s + (0.500 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.319 − 0.553i)6-s + (−0.290 + 0.503i)7-s + 3·8-s − 2.59·9-s + (0.499 + 0.866i)10-s + (−2.94 + 5.10i)11-s + (0.319 + 0.553i)12-s + (2.08 + 3.61i)13-s + (0.290 + 0.503i)14-s + (−0.319 + 0.553i)15-s + (0.500 − 0.866i)16-s + (−1.92 + 3.34i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + 0.368·3-s + (0.250 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.130 − 0.225i)6-s + (−0.109 + 0.190i)7-s + 1.06·8-s − 0.864·9-s + (0.158 + 0.273i)10-s + (−0.888 + 1.53i)11-s + (0.0921 + 0.159i)12-s + (0.578 + 1.00i)13-s + (0.0777 + 0.134i)14-s + (−0.0824 + 0.142i)15-s + (0.125 − 0.216i)16-s + (−0.467 + 0.810i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.513 - 0.858i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.513 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60191 + 0.908121i\)
\(L(\frac12)\) \(\approx\) \(1.60191 + 0.908121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
151 \( 1 + (-3.75 + 11.7i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 0.638T + 3T^{2} \)
7 \( 1 + (0.290 - 0.503i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.94 - 5.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.08 - 3.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.92 - 3.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
23 \( 1 + (-4.37 + 7.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.628T + 29T^{2} \)
31 \( 1 + (-2.92 + 5.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.34 + 4.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.943T + 41T^{2} \)
43 \( 1 + (-5.14 - 8.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.809 + 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.839T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + (1.76 + 3.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 3.68T + 67T^{2} \)
71 \( 1 + (-0.586 - 1.01i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (3.10 + 5.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.697 + 1.20i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.85455743463940689537050382891, −9.754159315495910628863754077646, −8.682563565863391682182813155414, −7.982211233141814988365826229516, −7.04493320334213112817452961109, −6.20059044918384998016062309679, −4.64411237814365600127530892704, −3.99779065653292042402051144583, −2.64629856945892124005802924585, −2.17092578985876445345542388482, 0.78002682056977162374326259959, 2.67430615369067870616178550056, 3.65851773083065988216867815705, 5.24873943867558446408041025009, 5.53789787410967726040851779866, 6.59693868327683143133458432722, 7.68881214327781451278892522776, 8.354895547540611057151393009111, 9.102015335103533871718817838030, 10.38420103969121286000890431513

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