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Properties

Label 2-980-5.3-c0-0-0
Degree $2$
Conductor $980$
Sign $0.973 - 0.229i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s − 11-s + (0.707 + 0.707i)13-s + 1.00·15-s + (0.707 − 0.707i)17-s + 1.41i·19-s + (−1 − i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (−0.707 − 0.707i)33-s + (−1 + i)37-s + 1.00i·39-s + (−1 − i)43-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s − 11-s + (0.707 + 0.707i)13-s + 1.00·15-s + (0.707 − 0.707i)17-s + 1.41i·19-s + (−1 − i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (−0.707 − 0.707i)33-s + (−1 + i)37-s + 1.00i·39-s + (−1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377021900\)
\(L(\frac12)\) \(\approx\) \(1.377021900\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
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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

−10.01199000452250153496901768596, −9.533623839467004718998000396136, −8.470444193453749913417840227577, −8.236784217354965680177753195622, −6.78252651089909879221005121327, −5.79361056681761986120264298205, −4.94594334465607589755504266419, −3.96789358529827480857973350759, −2.97169001499720657383057040768, −1.66660624681013410520308477510, 1.73565657595194071449317819409, 2.67292826717381102709188148376, 3.51859022295874098333628347250, 5.16855975458253997647170586219, 5.93463645754754882786196146325, 6.91406619680705215927694020682, 7.77442704149977805765625909133, 8.261023208228835659818471109110, 9.338979494277823464893504871589, 10.24962124627074032104438149219

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