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Properties

Label 2-980-140.139-c1-0-30
Degree $2$
Conductor $980$
Sign $-0.206 - 0.978i$
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 + (−1.73 + 2.64i)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 + (−0.549 + 0.835i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.206 - 0.978i$
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: \((2,\ 980,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47069 + 1.81329i\)
\(L(\frac12)\) \(\approx\) \(1.47069 + 1.81329i\)
\(L(\frac{3}{2})\) 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

−10.25044911313293856488680672593, −9.422083344440308807724647626924, −8.050881579850480061489194841235, −7.42540235601784689150584550247, −6.77370063842249558271217520304, −6.36071163432435031720456734047, −4.96372902290262285551229615499, −4.18884539763060867832554047917, −2.78980971191265311540899910876, −2.02134258650452765597460491954, 0.825828449723787310694362672260, 2.40527950664686895965153624787, 3.67219821074116647303507570455, 4.40063816118343567452818661534, 5.15304697045377402762981261006, 5.90258789872149738143769009450, 7.04913345054060719718509125250, 8.376659648928959368317311451859, 9.287328764027595688252145073942, 9.707014206296338958331955654332

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