↑

Properties

Label 2-605-55.9-c1-0-16
Degree $2$
Conductor $605$
Sign $-0.763 + 0.645i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (−1.92 − 0.625i)2-s + (−1.71 − 2.35i)3-s + (1.69 + 1.22i)4-s + (2.16 + 0.560i)5-s + (1.82 + 5.60i)6-s + (−1.88 + 2.59i)7-s + (−0.109 − 0.150i)8-s + (−1.69 + 5.22i)9-s + (−3.81 − 2.43i)10-s − 6.09i·12-s + (−0.617 − 0.200i)13-s + (5.25 − 3.81i)14-s + (−2.38 − 6.06i)15-s + (−1.17 − 3.62i)16-s + (1.13 − 0.367i)17-s + (6.53 − 8.99i)18-s + ⋯
L(s)  = 1  + (−1.36 − 0.441i)2-s + (−0.989 − 1.36i)3-s + (0.846 + 0.614i)4-s + (0.968 + 0.250i)5-s + (0.743 + 2.28i)6-s + (−0.713 + 0.981i)7-s + (−0.0385 − 0.0530i)8-s + (−0.566 + 1.74i)9-s + (−1.20 − 0.768i)10-s − 1.76i·12-s + (−0.171 − 0.0556i)13-s + (1.40 − 1.02i)14-s + (−0.616 − 1.56i)15-s + (−0.294 − 0.905i)16-s + (0.274 − 0.0890i)17-s + (1.54 − 2.11i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.763 + 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147288 - 0.402153i\)
\(L(\frac12)\) \(\approx\) \(0.147288 - 0.402153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.16 - 0.560i)T \)
11 \( 1 \)
good2 \( 1 + (1.92 + 0.625i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.71 + 2.35i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.88 - 2.59i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.617 + 0.200i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.13 + 0.367i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.52 + 1.11i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + (3.36 + 2.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.44 + 7.51i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.20 - 1.66i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.67 + 1.94i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (1.71 + 2.35i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.10 - 1.98i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.00 + 2.18i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.40 + 7.40i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 + (-4.85 - 14.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.35 + 4.62i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.71 + 14.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (10.1 - 3.28i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-14.3 - 4.67i)T + (78.4 + 57.0i)T^{2} \)
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

−10.25391309697558119065491954361, −9.509409938593905241765221451032, −8.699776783845252988029990373791, −7.65326524870441571161324744973, −6.80969379802521854339188375761, −6.00429308032911262388382770670, −5.28007926125618420945548975827, −2.62722291618291015738787479293, −1.92811765085835294357952619719, −0.50133605279959234878512982699, 1.12159001500092706094647477680, 3.49192285889044565670756349615, 4.64771597870965052776939641414, 5.70196794262876007472686692784, 6.51297216876947135204841100068, 7.42468040154356703920000416463, 8.755825984678434403009796106900, 9.621129566836387541529410421573, 9.887633995889794328512941399803, 10.54326197012703949336541663812

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