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Properties

Label 2-980-140.123-c1-0-85
Degree $2$
Conductor $980$
Sign $-0.382 + 0.924i$
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  + (−0.977 + 1.02i)2-s + (−0.346 − 1.29i)3-s + (−0.0888 − 1.99i)4-s + (1.92 + 1.13i)5-s + (1.65 + 0.909i)6-s + (2.12 + 1.86i)8-s + (1.04 − 0.605i)9-s + (−3.04 + 0.855i)10-s + (−4.33 − 2.50i)11-s + (−2.55 + 0.806i)12-s + (−3.27 − 3.27i)13-s + (0.802 − 2.88i)15-s + (−3.98 + 0.355i)16-s + (−0.873 − 3.25i)17-s + (−0.406 + 1.66i)18-s + (0.214 + 0.371i)19-s + ⋯
L(s)  = 1  + (−0.691 + 0.722i)2-s + (−0.199 − 0.745i)3-s + (−0.0444 − 0.999i)4-s + (0.861 + 0.508i)5-s + (0.677 + 0.371i)6-s + (0.752 + 0.658i)8-s + (0.349 − 0.201i)9-s + (−0.962 + 0.270i)10-s + (−1.30 − 0.754i)11-s + (−0.736 + 0.232i)12-s + (−0.908 − 0.908i)13-s + (0.207 − 0.743i)15-s + (−0.996 + 0.0887i)16-s + (−0.211 − 0.790i)17-s + (−0.0957 + 0.392i)18-s + (0.0491 + 0.0851i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.382 + 0.924i$
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: \((2,\ 980,\ (\ :1/2),\ -0.382 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360001 - 0.538555i\)
\(L(\frac12)\) \(\approx\) \(0.360001 - 0.538555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.977 - 1.02i)T \)
5 \( 1 + (-1.92 - 1.13i)T \)
7 \( 1 \)
good3 \( 1 + (0.346 + 1.29i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (4.33 + 2.50i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.27 + 3.27i)T + 13iT^{2} \)
17 \( 1 + (0.873 + 3.25i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.214 - 0.371i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.47 + 0.931i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.51iT - 29T^{2} \)
31 \( 1 + (0.353 + 0.203i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.33 + 1.16i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + (4.01 - 4.01i)T - 43iT^{2} \)
47 \( 1 + (-2.60 + 9.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.10 + 0.296i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.93 + 3.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.11 + 10.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.3 - 3.03i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (5.17 - 1.38i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.0987 - 0.171i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.86 - 1.86i)T - 83iT^{2} \)
89 \( 1 + (-6.11 + 3.52i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.24 - 5.24i)T - 97iT^{2} \)
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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

−9.803673795469768490787268136212, −8.837439508269826176470153593153, −7.79930010620495554118540700900, −7.29145880946703747190787432466, −6.47158933757956989163347902793, −5.62942902076226809021364486852, −5.00322601057749533607576349580, −2.96094102274974801282066261109, −1.88218677414507667079037440762, −0.36134972982272707366935924676, 1.77029164490799982217354865953, 2.53661583987058556014638709004, 4.22019630156507925985326002018, 4.69318842452883038961168064273, 5.81159299617670129439764104307, 7.16053333898714722876077212580, 7.903893255597771659471519672822, 8.955650454164591086107609450534, 9.675163733149239924978824176861, 10.17990704824287686019526051376

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