↑

Properties

Label 2-605-55.13-c1-0-16
Degree $2$
Conductor $605$
Sign $0.656 + 0.754i$
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  + (−2.16 + 1.10i)2-s + (−0.590 − 0.0935i)3-s + (2.30 − 3.16i)4-s + (−2.23 − 0.0720i)5-s + (1.38 − 0.449i)6-s + (0.403 + 2.54i)7-s + (−0.729 + 4.60i)8-s + (−2.51 − 0.816i)9-s + (4.92 − 2.31i)10-s + (−1.65 + 1.65i)12-s + (1.27 + 2.50i)13-s + (−3.69 − 5.08i)14-s + (1.31 + 0.251i)15-s + (−1.08 − 3.33i)16-s + (−0.428 + 0.841i)17-s + (6.34 − 1.00i)18-s + ⋯
L(s)  = 1  + (−1.53 + 0.780i)2-s + (−0.341 − 0.0540i)3-s + (1.15 − 1.58i)4-s + (−0.999 − 0.0322i)5-s + (0.564 − 0.183i)6-s + (0.152 + 0.963i)7-s + (−0.257 + 1.62i)8-s + (−0.837 − 0.272i)9-s + (1.55 − 0.731i)10-s + (−0.478 + 0.478i)12-s + (0.353 + 0.694i)13-s + (−0.986 − 1.35i)14-s + (0.339 + 0.0649i)15-s + (−0.271 − 0.834i)16-s + (−0.103 + 0.204i)17-s + (1.49 − 0.237i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.656 + 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180127 - 0.0819800i\)
\(L(\frac12)\) \(\approx\) \(0.180127 - 0.0819800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.0720i)T \)
11 \( 1 \)
good2 \( 1 + (2.16 - 1.10i)T + (1.17 - 1.61i)T^{2} \)
3 \( 1 + (0.590 + 0.0935i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.403 - 2.54i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (-1.27 - 2.50i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.428 - 0.841i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.35 - 2.43i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.104 - 0.104i)T + 23iT^{2} \)
29 \( 1 + (5.61 + 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.80 + 8.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.50 - 1.18i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.02 + 2.78i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.91 + 3.91i)T - 43iT^{2} \)
47 \( 1 + (-0.165 + 1.04i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.67 + 1.87i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-3.80 + 5.24i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.05 + 2.94i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \)
71 \( 1 + (-0.391 - 1.20i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.74 - 0.277i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (-1.34 + 4.12i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.38 + 3.76i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + 4.23iT - 89T^{2} \)
97 \( 1 + (6.54 + 12.8i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.40892195343059916880556669122, −9.310239544529941024870192975568, −8.601291046660323903267800613629, −8.183563698842311799047467068276, −7.12755419280861443248119343011, −6.24055255181934937558037937648, −5.49840750732569676512148700115, −3.89886660492053841537289239889, −2.10562108786635687105432130139, −0.23875061261222933360742975540, 0.998823023284680389987771168921, 2.75888175493507979121116037053, 3.78248169289077995207966084712, 5.14362386829682844302414466391, 6.80571399262661554224971513243, 7.50818334588515286827467120438, 8.417654613844193241842067211951, 8.839217844916822736145244802436, 10.19078664711016548906800654431, 10.83224519272339924466844290188

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