Properties

Label 2-980-35.9-c1-0-14
Degree $2$
Conductor $980$
Sign $0.308 + 0.951i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.5i)3-s + (−1.86 + 1.23i)5-s + (3 − 5.19i)9-s + (−1.5 − 2.59i)11-s + i·13-s + (−3 + 6i)15-s + (4.33 − 2.5i)17-s + (4 − 6.92i)19-s + (1.73 + i)23-s + (1.96 − 4.59i)25-s − 9i·27-s + 29-s + (−1 − 1.73i)31-s + (−7.79 − 4.5i)33-s + (−8.66 − 5i)37-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (−0.834 + 0.550i)5-s + (1 − 1.73i)9-s + (−0.452 − 0.783i)11-s + 0.277i·13-s + (−0.774 + 1.54i)15-s + (1.05 − 0.606i)17-s + (0.917 − 1.58i)19-s + (0.361 + 0.208i)23-s + (0.392 − 0.919i)25-s − 1.73i·27-s + 0.185·29-s + (−0.179 − 0.311i)31-s + (−1.35 − 0.783i)33-s + (−1.42 − 0.821i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.308 + 0.951i$
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: \((2,\ 980,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84938 - 1.34465i\)
\(L(\frac12)\) \(\approx\) \(1.84938 - 1.34465i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (-4.33 + 2.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.66 + 5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.5 - 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.498201409521387193095403759754, −8.883375524921561574878581081496, −8.058628622037710036541747394354, −7.33424384162469435487554391358, −7.00048018237222261760213757137, −5.59450076491160364774826823158, −4.15573363442531655335855105067, −3.04920154635995798618464664738, −2.72016682766894819974051636666, −0.966976187334345446841872998318, 1.68824620037295453690926620688, 3.16506432793313423237778978685, 3.71608156313355205832902244823, 4.66841974426817634879557763709, 5.52920260136622659280937084706, 7.31334669445838623682639990113, 7.86823385909909707634155860447, 8.473582595800703477796539695674, 9.255980668023784122185753333510, 10.13733627562758213459076830870

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