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Properties

Label 2-980-35.18-c0-0-0
Degree $2$
Conductor $980$
Sign $0.849 - 0.527i$
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  + (−0.965 + 0.258i)3-s + (−0.965 − 0.258i)5-s + (0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + 15-s + (0.258 + 0.965i)17-s + (1.22 − 0.707i)19-s + (−0.366 + 1.36i)23-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + i·29-s + (−0.258 + 0.965i)33-s + (1.36 + 0.366i)37-s + (−0.866 − 0.500i)39-s + (−1 − i)43-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.965 − 0.258i)5-s + (0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + 15-s + (0.258 + 0.965i)17-s + (1.22 − 0.707i)19-s + (−0.366 + 1.36i)23-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + i·29-s + (−0.258 + 0.965i)33-s + (1.36 + 0.366i)37-s + (−0.866 − 0.500i)39-s + (−1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6311925355\)
\(L(\frac12)\) \(\approx\) \(0.6311925355\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−10.56286511294802651320947465021, −9.378426673711768055384137840163, −8.630299639871096628452724969362, −7.80358593249647091562027413316, −6.78932905422992200252716087215, −5.88320990027071205794112932130, −5.14267329275528708769090060861, −4.06114031359897918382081830182, −3.28247636002593988640477560388, −1.17480259961809420165582947735, 0.880639010377330123022044505635, 2.82727255547351122793953757457, 3.95393156137740829287715055653, 4.92486522426650311253599991176, 5.92695738461374144117486568788, 6.69422356873869319614693522437, 7.56171693189132966059616828099, 8.248441078673464379149040908010, 9.413367900884121490498917104210, 10.27483464936596689688323837473

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