↑

Properties

Label 2-350-35.23-c2-0-23
Degree $2$
Conductor $350$
Sign $-0.756 - 0.653i$
Analytic cond. $9.53680$
Root an. cond. $3.08817$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 1.36i)2-s + (1.31 − 4.90i)3-s + (−1.73 + i)4-s − 7.18·6-s + (0.145 − 6.99i)7-s + (2 + 1.99i)8-s + (−14.5 − 8.41i)9-s + (0.474 + 0.821i)11-s + (2.63 + 9.81i)12-s + (−0.862 − 0.862i)13-s + (−9.61 + 2.36i)14-s + (1.99 − 3.46i)16-s + (−29.3 − 7.87i)17-s + (−6.15 + 22.9i)18-s + (20.9 + 12.0i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.438 − 1.63i)3-s + (−0.433 + 0.250i)4-s − 1.19·6-s + (0.0207 − 0.999i)7-s + (0.250 + 0.249i)8-s + (−1.61 − 0.934i)9-s + (0.0431 + 0.0746i)11-s + (0.219 + 0.818i)12-s + (−0.0663 − 0.0663i)13-s + (−0.686 + 0.168i)14-s + (0.124 − 0.216i)16-s + (−1.72 − 0.463i)17-s + (−0.342 + 1.27i)18-s + (1.10 + 0.635i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.756 - 0.653i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.756 - 0.653i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.435460 + 1.16969i\)
\(L(\frac12)\) \(\approx\) \(0.435460 + 1.16969i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-0.145 + 6.99i)T \)
good3 \( 1 + (-1.31 + 4.90i)T + (-7.79 - 4.5i)T^{2} \)
11 \( 1 + (-0.474 - 0.821i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.862 + 0.862i)T + 169iT^{2} \)
17 \( 1 + (29.3 + 7.87i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-20.9 - 12.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.47 + 1.46i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 7.33iT - 841T^{2} \)
31 \( 1 + (23.5 + 40.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-2.13 - 7.95i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 53.3T + 1.68e3T^{2} \)
43 \( 1 + (-33.0 - 33.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-7.87 - 29.3i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (3.48 - 12.9i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-43.5 + 25.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.7 + 39.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (65.9 + 17.6i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 + (-14.2 + 53.2i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (61.4 + 35.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (85.7 + 85.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (135. + 78.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-87.1 + 87.1i)T - 9.40e3iT^{2} \)
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

−11.02218023573195915630753174228, −9.698704752467886212135972831409, −8.792145672264926477000119715827, −7.67025793652726945043091205965, −7.23574657319895670232015637132, −6.07631262155311616315678807352, −4.36893214800503334121464629813, −2.97114079215687035321242641359, −1.78966940463849846455120417345, −0.57433651463646520564403538208, 2.60659435539921888578166445346, 3.95939612029259174173467669341, 4.95637194429139418522586393608, 5.75778385755069325457300180699, 7.11758165473466764107962810313, 8.617282120203932173259913976327, 8.927750459130512518464007895511, 9.706700693418675624658603492536, 10.73801702997939102841159050106, 11.48999998270733236404379902500

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