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Properties

Label 2-980-7.4-c1-0-0
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  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s − 13-s + 0.999·15-s + (1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s − 5·27-s − 9·29-s + (−4 + 6.92i)31-s + (−1.5 − 2.59i)33-s + (5 + 8.66i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s − 0.277·13-s + 0.258·15-s + (0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s − 1.67·29-s + (−0.718 + 1.24i)31-s + (−0.261 − 0.452i)33-s + (0.821 + 1.42i)37-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

\(L(1)\) \(\approx\) \(0.390873 + 0.788485i\)
\(L(\frac12)\) \(\approx\) \(0.390873 + 0.788485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−10.19878233630553201070847709145, −9.584924796427693982190559879827, −8.777127298541232457126279446556, −7.50389391736109657239854899061, −7.28627167712099794396599284422, −5.72011459006091455358417945390, −4.98870038823146990883262032440, −4.32970782677682468646644018440, −3.04149919230460280942316065031, −1.63147896362715371417168117469, 0.41746101808479425691552841503, 2.02138164466295000112059816738, 3.34791081269071475089753149529, 4.23875086691898894471816145757, 5.68524353764833429902033358503, 6.18359118117925877405878665415, 7.27998785116000433031061430270, 7.80463771312474312150857378387, 8.889497265194187587586100394818, 9.695375697406569904243552067044

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