↑

Properties

Label 2-350-35.9-c3-0-1
Degree $2$
Conductor $350$
Sign $-0.951 + 0.308i$
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 + (−6.06 + 17.5i)7-s − 7.99i·8-s + (−13 + 22.5i)9-s + (15 + 25.9i)11-s + (−3.46 − 1.99i)12-s − 44i·13-s + (28 − 24.2i)14-s + (−8 + 13.8i)16-s + (−20.7 + 12i)17-s + (45.0 − 26i)18-s + (1 − 1.73i)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 + (−0.327 + 0.944i)7-s − 0.353i·8-s + (−0.481 + 0.833i)9-s + (0.411 + 0.712i)11-s + (−0.0833 − 0.0481i)12-s − 0.938i·13-s + (0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.296 + 0.171i)17-s + (0.589 − 0.340i)18-s + (0.0120 − 0.0209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.951 + 0.308i$
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: \((2,\ 350,\ (\ :3/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.01470295728\)
\(L(\frac12)\) \(\approx\) \(0.01470295728\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.06 - 17.5i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 44iT - 2.19e3T^{2} \)
17 \( 1 + (20.7 - 12i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (158. + 91.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 279T + 2.43e4T^{2} \)
31 \( 1 + (-20 - 34.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-65.8 - 38i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 423T + 6.89e4T^{2} \)
43 \( 1 + 305iT - 7.95e4T^{2} \)
47 \( 1 + (394. + 228i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-171. + 99i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (231 + 400. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (140.5 - 243. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (432. - 249.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 534T + 3.57e5T^{2} \)
73 \( 1 + (692. - 400i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (395 - 684. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 597iT - 5.71e5T^{2} \)
89 \( 1 + (-508.5 + 880. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3iT - 9.12e5T^{2} \)
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

−11.65338339315355573148793808365, −10.35758744285382114067196468631, −9.998417088175610015048846525296, −8.611737048678385451667748439911, −8.219967902367016902711952519234, −6.83300058533139844624165063592, −5.78959900953251138252565639059, −4.61580192639884810504546097873, −3.00529243991385617126636311801, −1.96827720886453684147924879592, 0.00666378425988742350005069743, 1.32078012904637386713302060655, 3.24779488451092730507876734323, 4.47326211035385029203959776051, 6.13702258971716133770925263041, 6.56616643264679743711495949865, 7.70993234080334128618081580624, 8.732564811417559925298590510541, 9.561676484759884944506523506802, 10.40128293711830140214430417385

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