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Properties

Label 2-980-35.9-c1-0-1
Degree $2$
Conductor $980$
Sign $-0.795 - 0.605i$
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  + (−2.95 + 1.70i)3-s + (−1.93 − 1.11i)5-s + (4.31 − 7.46i)9-s + (1.81 + 3.13i)11-s + 1.06i·13-s + 7.62·15-s + (4.98 − 2.87i)17-s + (2.5 + 4.33i)25-s + 19.1i·27-s − 9.62·29-s + (−10.6 − 6.17i)33-s + (−1.81 − 3.13i)39-s + (−16.6 + 9.64i)45-s + (1.11 + 0.641i)47-s + (−9.81 + 16.9i)51-s + ⋯
L(s)  = 1  + (−1.70 + 0.984i)3-s + (−0.866 − 0.499i)5-s + (1.43 − 2.48i)9-s + (0.546 + 0.946i)11-s + 0.294i·13-s + 1.96·15-s + (1.20 − 0.697i)17-s + (0.5 + 0.866i)25-s + 3.68i·27-s − 1.78·29-s + (−1.86 − 1.07i)33-s + (−0.290 − 0.502i)39-s + (−2.48 + 1.43i)45-s + (0.162 + 0.0936i)47-s + (−1.37 + 2.37i)51-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.795 - 0.605i$
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: \((2,\ 980,\ (\ :1/2),\ -0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132644 + 0.393437i\)
\(L(\frac12)\) \(\approx\) \(0.132644 + 0.393437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (2.95 - 1.70i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.81 - 3.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.06iT - 13T^{2} \)
17 \( 1 + (-4.98 + 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.62T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-1.11 - 0.641i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (11.6 - 6.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.43 - 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.3iT - 97T^{2} \)
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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

−10.36081407522600548713894541913, −9.621097181352916969904605970696, −9.049172846104238787204296347756, −7.54695493120405481321484852289, −6.91412949341421074406220495656, −5.77491385540511643629230300328, −5.07931226721340357591399541493, −4.28896022136375303603955783718, −3.60850672815947751575776469480, −1.15997634819695405513927187988, 0.29663885190631633339350597265, 1.56550925293256038233080636847, 3.34215378510733115021583407799, 4.49238078660699266003052792034, 5.76320636138507167471413539321, 6.03706312222840944212848760187, 7.20179029309873858164768915372, 7.60210568675166169125319289658, 8.561618333365494954077500368114, 10.10174684667051146792269063084

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