↑

Properties

Label 2-980-35.9-c1-0-3
Degree $2$
Conductor $980$
Sign $-0.911 - 0.412i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
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  + (−1.5 + 0.866i)3-s + (0.5 + 2.17i)5-s + (2.63 + 4.56i)11-s − 2.62i·13-s + (−2.63 − 2.83i)15-s + (0.362 − 0.209i)17-s + (−1.63 + 2.83i)19-s + (6.77 + 3.91i)23-s + (−4.50 + 2.17i)25-s − 5.19i·27-s − 4.27·29-s + (1.63 + 2.83i)31-s + (−7.91 − 4.56i)33-s + (−8.63 − 4.98i)37-s + (2.27 + 3.94i)39-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.223 + 0.974i)5-s + (0.795 + 1.37i)11-s − 0.728i·13-s + (−0.680 − 0.732i)15-s + (0.0879 − 0.0507i)17-s + (−0.375 + 0.650i)19-s + (1.41 + 0.815i)23-s + (−0.900 + 0.435i)25-s − 0.999i·27-s − 0.793·29-s + (0.294 + 0.509i)31-s + (−1.37 − 0.795i)33-s + (−1.41 − 0.819i)37-s + (0.364 + 0.630i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.911 - 0.412i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.911 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194818 + 0.903310i\)
\(L(\frac12)\) \(\approx\) \(0.194818 + 0.903310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 - 2.17i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + (-0.362 + 0.209i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 - 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.63 + 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 + 2.15iT - 43T^{2} \)
47 \( 1 + (5.63 + 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 - 2.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.77 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.04 - 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 + (5.63 - 3.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.63 + 6.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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

−10.39774230944434407722982435869, −9.827209897010742525900501684751, −8.898194380927100571395902447303, −7.54275153020867288793703942787, −6.99305180432621922031995664618, −5.97920303257607377786024711994, −5.26111938390530634342607480321, −4.24501330327971397060252489428, −3.15461251450339561803003732905, −1.78626766896033821206682651831, 0.49432236018666601283806348925, 1.54754697755121630021168138326, 3.27161885659985407329080877494, 4.52643281564899588297211304365, 5.37218873161892973176634115012, 6.30207563323264042742466128338, 6.73641738802970493079039138657, 8.079297764924368177916192108257, 8.979407985505011949424405473985, 9.311228655490368442180356890967

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