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Properties

Label 2-980-28.27-c1-0-71
Degree $2$
Conductor $980$
Sign $-0.988 + 0.153i$
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  + (0.486 − 1.32i)2-s + 0.812·3-s + (−1.52 − 1.29i)4-s + i·5-s + (0.395 − 1.07i)6-s + (−2.45 + 1.39i)8-s − 2.34·9-s + (1.32 + 0.486i)10-s − 4.86i·11-s + (−1.23 − 1.04i)12-s − 0.895i·13-s + 0.812i·15-s + (0.658 + 3.94i)16-s − 5.89i·17-s + (−1.13 + 3.10i)18-s + 2.91·19-s + ⋯
L(s)  = 1  + (0.344 − 0.938i)2-s + 0.468·3-s + (−0.763 − 0.646i)4-s + 0.447i·5-s + (0.161 − 0.440i)6-s + (−0.869 + 0.494i)8-s − 0.780·9-s + (0.419 + 0.153i)10-s − 1.46i·11-s + (−0.357 − 0.302i)12-s − 0.248i·13-s + 0.209i·15-s + (0.164 + 0.986i)16-s − 1.42i·17-s + (−0.268 + 0.732i)18-s + 0.669·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100747 - 1.30258i\)
\(L(\frac12)\) \(\approx\) \(0.100747 - 1.30258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.486 + 1.32i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 0.812T + 3T^{2} \)
11 \( 1 + 4.86iT - 11T^{2} \)
13 \( 1 + 0.895iT - 13T^{2} \)
17 \( 1 + 5.89iT - 17T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 + 1.56iT - 23T^{2} \)
29 \( 1 + 9.73T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
37 \( 1 + 1.82T + 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 - 3.04iT - 43T^{2} \)
47 \( 1 - 5.21T + 47T^{2} \)
53 \( 1 - 0.179T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 6.15iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 - 8.38iT - 71T^{2} \)
73 \( 1 - 8.78iT - 73T^{2} \)
79 \( 1 - 3.63iT - 79T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 - 1.76iT - 89T^{2} \)
97 \( 1 - 7.83iT - 97T^{2} \)
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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

−9.540039609855006429227133165902, −9.010890231600696618561209086495, −8.184885629430102933150142795600, −7.11670574166279779450310917356, −5.73672960403112726777708701634, −5.36081671228079540217734441295, −3.76769882447282725850330191717, −3.15988627730599665660278973550, −2.28550498982775698709388956719, −0.48864877337221501100155742563, 1.94827766752038157008779945988, 3.44953260928586676375444226779, 4.27510712001831073424889201829, 5.31469127625601673334809850527, 6.01944274207922397667575317825, 7.20196041744080815929360081510, 7.76294073157571959043885257663, 8.671217191108684932740849686616, 9.283397064106007892830599991296, 10.04359702330412623873504782777

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