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Properties

Label 2-980-7.4-c1-0-9
Degree $2$
Conductor $980$
Sign $0.605 + 0.795i$
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 − 1.73i)3-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + 2·13-s + 1.99·15-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + 4.00·27-s + 6·29-s + (2 − 3.46i)31-s + (−1 − 1.73i)37-s + (2 − 3.46i)39-s + 6·41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + (−0.166 − 0.288i)9-s + 0.554·13-s + 0.516·15-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + 0.769·27-s + 1.11·29-s + (0.359 − 0.622i)31-s + (−0.164 − 0.284i)37-s + (0.320 − 0.554i)39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91288 - 0.948268i\)
\(L(\frac12)\) \(\approx\) \(1.91288 - 0.948268i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.924094709902073761501248050588, −8.889299820742350337348374369779, −8.053025319550978287116404368811, −7.45324651381589791605491044968, −6.59858366509367480952496458667, −5.79616924998664611328516655974, −4.55247514025862520893648744209, −3.20179567324733865279262650438, −2.36914283912356409762599381318, −1.11404015255516090492210452155, 1.41411687235421623832430128761, 3.01896128708162683851763325716, 3.82310851333526498030825527579, 4.72867288078326871471190934112, 5.67983218297802034433494315335, 6.65459754637677279776876050756, 7.913492478369081500591229610837, 8.609180564786340551572534074102, 9.308670066528639677666537036848, 10.08148831011459137485763202455

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