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Properties

Label 2-980-5.4-c1-0-18
Degree $2$
Conductor $980$
Sign $-0.894 + 0.447i$
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  − 3i·3-s + (2 − i)5-s − 6·9-s + 3·11-s − i·13-s + (−3 − 6i)15-s − 5i·17-s − 8·19-s + 2i·23-s + (3 − 4i)25-s + 9i·27-s + 29-s + 2·31-s − 9i·33-s − 10i·37-s + ⋯
L(s)  = 1  − 1.73i·3-s + (0.894 − 0.447i)5-s − 2·9-s + 0.904·11-s − 0.277i·13-s + (−0.774 − 1.54i)15-s − 1.21i·17-s − 1.83·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s + 1.73i·27-s + 0.185·29-s + 0.359·31-s − 1.56i·33-s − 1.64i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384578 - 1.62909i\)
\(L(\frac12)\) \(\approx\) \(0.384578 - 1.62909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
7 \( 1 \)
good3 \( 1 + 3iT - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 11iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 7T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 3iT - 97T^{2} \)
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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

−9.308647782440550662308877980361, −8.839989837474824533469769095291, −7.85331663001520225087384247214, −7.03991450957754779616423359108, −6.27392318827859543239492030933, −5.70687099983519711550228008734, −4.41093566146988608438378054487, −2.71340007281772235735847648629, −1.86820668411512967594478598834, −0.77182193178669602856374884423, 2.01841951785734878764055362974, 3.31062373178681191280085470489, 4.18923555886163601576632259182, 4.92756198997538272549689692150, 6.17872031279806468540302374260, 6.48267042328027339439777400728, 8.261941328595009393414374843364, 8.935894304999737881634882744772, 9.622560021151432896492148975476, 10.41335474582043844198113590174

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