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Properties

Label 2-980-140.123-c1-0-75
Degree $2$
Conductor $980$
Sign $0.975 + 0.221i$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (−0.133 − 2.23i)5-s + (1.99 + 2i)8-s + (2.59 − 1.5i)9-s + (0.633 − 3.09i)10-s + (1 + i)13-s + (1.99 + 3.46i)16-s + (−1.09 − 4.09i)17-s + (4.09 − 1.09i)18-s + (2 − 3.99i)20-s + (−4.96 + 0.598i)25-s + (1 + 1.73i)26-s − 4i·29-s + (1.46 + 5.46i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (−0.0599 − 0.998i)5-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (0.200 − 0.979i)10-s + (0.277 + 0.277i)13-s + (0.499 + 0.866i)16-s + (−0.266 − 0.993i)17-s + (0.965 − 0.258i)18-s + (0.447 − 0.894i)20-s + (−0.992 + 0.119i)25-s + (0.196 + 0.339i)26-s − 0.742i·29-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11177 - 0.349234i\)
\(L(\frac12)\) \(\approx\) \(3.11177 - 0.349234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (1.09 + 4.09i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.56 - 2.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.2 - 3.29i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (15.0 - 4.02i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (13.8 - 8i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−9.850926665717516288592502923592, −9.170296706160873688643479177860, −8.105078409525401734287031782705, −7.35310095605237245086968015954, −6.42778427606911589056697556357, −5.57484291456888594466816570042, −4.50708507308186930885239646859, −4.09496297284931900555218036566, −2.68400506719094407886163835143, −1.25328371338526509910136097229, 1.64709144584983628216749916858, 2.74645250731402683034128553097, 3.77957750497708541029536593276, 4.56876047824753049778130665545, 5.77163018534727441130035604470, 6.48622191686268831445444489127, 7.32348334146606884167499512863, 8.050227197790419625948410463305, 9.548558860533494727017799527171, 10.32762166792068688061922336229

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