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Properties

Label 2-980-28.27-c1-0-68
Degree $2$
Conductor $980$
Sign $0.860 + 0.509i$
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  + (1.39 − 0.226i)2-s + 1.79·3-s + (1.89 − 0.632i)4-s − i·5-s + (2.49 − 0.405i)6-s + (2.50 − 1.31i)8-s + 0.206·9-s + (−0.226 − 1.39i)10-s + 4.23i·11-s + (3.39 − 1.13i)12-s − 2.98i·13-s − 1.79i·15-s + (3.19 − 2.40i)16-s − 2.21i·17-s + (0.288 − 0.0468i)18-s + 4.56·19-s + ⋯
L(s)  = 1  + (0.987 − 0.160i)2-s + 1.03·3-s + (0.948 − 0.316i)4-s − 0.447i·5-s + (1.02 − 0.165i)6-s + (0.885 − 0.464i)8-s + 0.0689·9-s + (−0.0716 − 0.441i)10-s + 1.27i·11-s + (0.980 − 0.327i)12-s − 0.827i·13-s − 0.462i·15-s + (0.799 − 0.600i)16-s − 0.537i·17-s + (0.0680 − 0.0110i)18-s + 1.04·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.860 + 0.509i$
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: \((2,\ 980,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.96824 - 1.08780i\)
\(L(\frac12)\) \(\approx\) \(3.96824 - 1.08780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.226i)T \)
5 \( 1 + iT \)
7 \( 1 \)
good3 \( 1 - 1.79T + 3T^{2} \)
11 \( 1 - 4.23iT - 11T^{2} \)
13 \( 1 + 2.98iT - 13T^{2} \)
17 \( 1 + 2.21iT - 17T^{2} \)
19 \( 1 - 4.56T + 19T^{2} \)
23 \( 1 - 2.05iT - 23T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + 2.40T + 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 + 4.88iT - 41T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + 6.76T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 + 0.0226iT - 61T^{2} \)
67 \( 1 - 5.06iT - 67T^{2} \)
71 \( 1 + 4.07iT - 71T^{2} \)
73 \( 1 + 3.33iT - 73T^{2} \)
79 \( 1 - 3.62iT - 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 16.6iT - 89T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−9.790949367706515258173616923898, −9.299626347242446577075661180804, −7.978878805386600797134541033374, −7.56775209278982203751034904757, −6.46936660953977215787787552530, −5.29018292342399297706462270067, −4.65862194989030031197992411903, −3.44940502504428906989669175258, −2.73764181455946898719105299961, −1.54981500922309939627542348029, 1.85830878008574147021592834847, 3.09038327291096125125521203664, 3.46145819099473033641962933345, 4.68448410517532051367686299802, 5.83457806615879956443027780755, 6.55687832398514942469006964213, 7.54232001329272183397345480185, 8.314549720793111438702563586632, 9.017365630627522458215342514510, 10.16764493346148005069741531365

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