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Properties

Label 2-980-140.139-c1-0-82
Degree $2$
Conductor $980$
Sign $-0.121 + 0.992i$
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  + (−1.14 + 0.826i)2-s − 1.52i·3-s + (0.632 − 1.89i)4-s + (−0.0967 − 2.23i)5-s + (1.25 + 1.74i)6-s + (0.843 + 2.69i)8-s + 0.679·9-s + (1.95 + 2.48i)10-s − 4.56i·11-s + (−2.89 − 0.963i)12-s + 2.19·13-s + (−3.40 + 0.147i)15-s + (−3.19 − 2.40i)16-s + 6.22·17-s + (−0.779 + 0.561i)18-s + 3.83·19-s + ⋯
L(s)  = 1  + (−0.811 + 0.584i)2-s − 0.879i·3-s + (0.316 − 0.948i)4-s + (−0.0432 − 0.999i)5-s + (0.514 + 0.713i)6-s + (0.298 + 0.954i)8-s + 0.226·9-s + (0.619 + 0.785i)10-s − 1.37i·11-s + (−0.834 − 0.278i)12-s + 0.608·13-s + (−0.878 + 0.0380i)15-s + (−0.799 − 0.600i)16-s + 1.50·17-s + (−0.183 + 0.132i)18-s + 0.879·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.121 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776329 - 0.876745i\)
\(L(\frac12)\) \(\approx\) \(0.776329 - 0.876745i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 0.826i)T \)
5 \( 1 + (0.0967 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + 1.52iT - 3T^{2} \)
11 \( 1 + 4.56iT - 11T^{2} \)
13 \( 1 - 2.19T + 13T^{2} \)
17 \( 1 - 6.22T + 17T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
23 \( 1 + 0.430T + 23T^{2} \)
29 \( 1 + 0.473T + 29T^{2} \)
31 \( 1 - 7.59T + 31T^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 - 4.48iT - 47T^{2} \)
53 \( 1 + 9.23iT - 53T^{2} \)
59 \( 1 + 3.13T + 59T^{2} \)
61 \( 1 + 5.71iT - 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 - 4.57iT - 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 6.20iT - 79T^{2} \)
83 \( 1 + 7.69iT - 83T^{2} \)
89 \( 1 - 9.32iT - 89T^{2} \)
97 \( 1 - 9.05T + 97T^{2} \)
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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

−9.682391874296089496082764474950, −8.608155002943498185702838174237, −8.153830461724776049936877191953, −7.50163418586898877953920142636, −6.37522773784357595387623650421, −5.77868607927422726594564130803, −4.82046219241006529684567920476, −3.23544486048738142122875553396, −1.46404277189057871627025883130, −0.835488557313237641531649055016, 1.56104652528124147837172521926, 2.94281697065359755195184396530, 3.71660610550204236706954530670, 4.66576978832121871004714049816, 6.05865009731841790933862598528, 7.29978133916480186271477505842, 7.56251309689087997602608676808, 8.845479828696171540135891688953, 9.812594384294393713424915408620, 10.08216269614710369615541885226

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