Properties

Label 2-980-5.4-c1-0-6
Degree $2$
Conductor $980$
Sign $0.447 - 0.894i$
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 + 2i)5-s + 3·9-s − 4i·13-s + 4i·17-s + 4·19-s + 8i·23-s + (−3 − 4i)25-s − 2·29-s + 8·31-s + 8i·37-s − 6·41-s + 8i·43-s + (−3 + 6i)45-s + 8i·47-s − 4·59-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + 9-s − 1.10i·13-s + 0.970i·17-s + 0.917·19-s + 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.371·29-s + 1.43·31-s + 1.31i·37-s − 0.937·41-s + 1.21i·43-s + (−0.447 + 0.894i)45-s + 1.16i·47-s − 0.520·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 980,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28325 + 0.793098i\)
\(L(\frac12)\) \(\approx\) \(1.28325 + 0.793098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−10.05233075025307437457255620931, −9.664454064208399134561197183584, −8.142734267157461736751636478283, −7.73245463829029371021455980558, −6.82286378000839713123443864296, −5.97249057170617375311967569198, −4.85222644296536460126174026308, −3.71245667115707378202967247638, −2.97490547179958909482585564847, −1.38123482634235965320934201771, 0.795800109119008986207509747722, 2.17543261314394075770418611888, 3.75797069881104069637612385180, 4.55558651605682777273395501259, 5.27094420487273348104079897818, 6.66267580602216179562967404843, 7.26938529790489518941486150390, 8.257272721551026313630727910791, 9.064975301251896972903238645405, 9.718041539332464626372441339389

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