↑

Properties

Label 2-350-35.9-c3-0-4
Degree $2$
Conductor $350$
Sign $-0.843 - 0.537i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−0.866 + 0.5i)3-s + (1.99 + 3.46i)4-s + 1.99·6-s + (15.5 + 10i)7-s − 7.99i·8-s + (−13 + 22.5i)9-s + (−17.5 − 30.3i)11-s + (−3.46 − 1.99i)12-s + 66i·13-s + (−17 − 32.9i)14-s + (−8 + 13.8i)16-s + (−51.0 + 29.5i)17-s + (45.0 − 26i)18-s + (68.5 − 118. i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.166 + 0.0962i)3-s + (0.249 + 0.433i)4-s + 0.136·6-s + (0.841 + 0.539i)7-s − 0.353i·8-s + (−0.481 + 0.833i)9-s + (−0.479 − 0.830i)11-s + (−0.0833 − 0.0481i)12-s + 1.40i·13-s + (−0.324 − 0.628i)14-s + (−0.125 + 0.216i)16-s + (−0.728 + 0.420i)17-s + (0.589 − 0.340i)18-s + (0.827 − 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.843 - 0.537i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4346159115\)
\(L(\frac12)\) \(\approx\) \(0.4346159115\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
5 \( 1 \)
7 \( 1 + (-15.5 - 10i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (17.5 + 30.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 66iT - 2.19e3T^{2} \)
17 \( 1 + (51.0 - 29.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.06 - 3.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 + 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-9.52 - 5.5i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 - 260iT - 7.95e4T^{2} \)
47 \( 1 + (148. + 85.5i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (361. - 208.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (8.5 + 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (25.5 - 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (380. - 219.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (-255. + 147.5i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (247.5 - 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 932iT - 5.71e5T^{2} \)
89 \( 1 + (436.5 - 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 290iT - 9.12e5T^{2} \)
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

−11.27780107227057520090023450805, −10.83501434834165295269195160591, −9.445243772292905876851597630195, −8.698131907035566336061597213125, −7.952461545165426288424942026521, −6.79342335330379276745001399767, −5.48627512588193070169632865529, −4.50684041782699093369725733523, −2.81685541331642997394762534438, −1.72047123469316578214081351328, 0.18648529725170281264397116484, 1.62178104421356867877333096098, 3.34373368019344576487904030684, 4.93501128887499179792932713009, 5.81267537121461483793939745436, 7.08961795491544387836212832849, 7.81303397676370054991861131247, 8.669951462840486619133822770637, 9.856956138536537690113929269290, 10.50123362936926461934863422141

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