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Properties

Label 2-980-35.9-c1-0-7
Degree $2$
Conductor $980$
Sign $0.982 + 0.185i$
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  + (−2.23 − 0.133i)5-s + (−1.5 + 2.59i)9-s − 4i·13-s + (3.46 − 2i)17-s + (2 − 3.46i)19-s + (6.92 + 4i)23-s + (4.96 + 0.598i)25-s − 2·29-s + (4 + 6.92i)31-s + (6.92 + 4i)37-s + 6·41-s − 8i·43-s + (3.69 − 5.59i)45-s + (−6.92 − 4i)47-s + (−2 − 3.46i)59-s + ⋯
L(s)  = 1  + (−0.998 − 0.0599i)5-s + (−0.5 + 0.866i)9-s − 1.10i·13-s + (0.840 − 0.485i)17-s + (0.458 − 0.794i)19-s + (1.44 + 0.834i)23-s + (0.992 + 0.119i)25-s − 0.371·29-s + (0.718 + 1.24i)31-s + (1.13 + 0.657i)37-s + 0.937·41-s − 1.21i·43-s + (0.550 − 0.834i)45-s + (−1.01 − 0.583i)47-s + (−0.260 − 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.982 + 0.185i$
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: \((2,\ 980,\ (\ :1/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26846 - 0.118757i\)
\(L(\frac12)\) \(\approx\) \(1.26846 - 0.118757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.133i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.92 + 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−10.03048080107623049725194204300, −9.050566794417287782413425083481, −8.128474708784298076304312557466, −7.64046123905680919581557778955, −6.77258980269466721507198350982, −5.29699790414913958053695680384, −4.96998853271478763571430683095, −3.47530894235330696219250094753, −2.78359223102205622521061339146, −0.845848652453346445710810900749, 0.962554427255944045818718986399, 2.78372263240156959442536848385, 3.77414100600725362662326140358, 4.53978395344829857762746738944, 5.84734254600871668037574197444, 6.63730428256926712623220216119, 7.58678863588756123840577605265, 8.310126054698841747483603129600, 9.199099342813155425546326455796, 9.881095382358240547002154553866

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