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Properties

Label 2-980-140.107-c1-0-108
Degree $2$
Conductor $980$
Sign $-0.952 - 0.305i$
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  + (−0.939 + 1.05i)2-s + (0.815 − 3.04i)3-s + (−0.235 − 1.98i)4-s + (0.727 − 2.11i)5-s + (2.45 + 3.71i)6-s + (2.32 + 1.61i)8-s + (−5.99 − 3.45i)9-s + (1.55 + 2.75i)10-s + (−3.44 + 1.98i)11-s + (−6.23 − 0.902i)12-s + (−2.52 + 2.52i)13-s + (−5.83 − 3.93i)15-s + (−3.88 + 0.935i)16-s + (−0.277 + 1.03i)17-s + (9.28 − 3.08i)18-s + (−0.658 + 1.13i)19-s + ⋯
L(s)  = 1  + (−0.664 + 0.747i)2-s + (0.470 − 1.75i)3-s + (−0.117 − 0.993i)4-s + (0.325 − 0.945i)5-s + (1.00 + 1.51i)6-s + (0.820 + 0.571i)8-s + (−1.99 − 1.15i)9-s + (0.490 + 0.871i)10-s + (−1.03 + 0.599i)11-s + (−1.79 − 0.260i)12-s + (−0.700 + 0.700i)13-s + (−1.50 − 1.01i)15-s + (−0.972 + 0.233i)16-s + (−0.0672 + 0.251i)17-s + (2.18 − 0.727i)18-s + (−0.150 + 0.261i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0813142 + 0.519910i\)
\(L(\frac12)\) \(\approx\) \(0.0813142 + 0.519910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 1.05i)T \)
5 \( 1 + (-0.727 + 2.11i)T \)
7 \( 1 \)
good3 \( 1 + (-0.815 + 3.04i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (3.44 - 1.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.52 - 2.52i)T - 13iT^{2} \)
17 \( 1 + (0.277 - 1.03i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.658 - 1.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.02 - 1.34i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.50iT - 29T^{2} \)
31 \( 1 + (2.23 - 1.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.29 + 1.95i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + (1.77 + 1.77i)T + 43iT^{2} \)
47 \( 1 + (1.76 + 6.60i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.08 + 2.16i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.17 + 7.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.673 + 1.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.32 + 1.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.47iT - 71T^{2} \)
73 \( 1 + (-2.33 - 0.624i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.72 + 2.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.37 + 8.37i)T + 83iT^{2} \)
89 \( 1 + (-8.15 - 4.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.04 + 9.04i)T + 97iT^{2} \)
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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

−9.332995195683657018038826366895, −8.377419764266753887942944994596, −7.84663595721786098900717400095, −7.28736854632979182318319705633, −6.27466576358365094135532938011, −5.63310640074719706596610911707, −4.48595546809435836143834098755, −2.31542142122861499604425147207, −1.71145228868262119213516711055, −0.26320723822096028111942144719, 2.60490723766874270010484035959, 2.90712901775413248007774851270, 3.98183504025865233645014490627, 4.95262525530108550490582187231, 6.01667998889665850322140056401, 7.60637653899251542410911694024, 8.122697219774289033439481579283, 9.244584057551915339083111749629, 9.687653562207903041479582759752, 10.54505484620921114837213181768

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