Properties

Label 2-980-140.39-c0-0-5
Degree $2$
Conductor $980$
Sign $0.328 + 0.944i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (−0.866 − 0.499i)25-s + (0.707 + 1.22i)26-s + (−0.866 − 0.499i)32-s − 1.41·34-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.258 − 0.965i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (−0.866 − 0.499i)25-s + (0.707 + 1.22i)26-s + (−0.866 − 0.499i)32-s − 1.41·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.698306124\)
\(L(\frac12)\) \(\approx\) \(1.698306124\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−10.09668256403529590553036571258, −9.339961940217215710522070140503, −8.590576557424563564510420489027, −7.26936529084338686246055238888, −6.53997326886289478204797337404, −5.39359275853674283116118357473, −4.65137667267739863832428856911, −4.08533714243369585529571767341, −2.43702674696240688646207824865, −1.56997585376123356412284055248, 2.20521203852142335097996614980, 3.31666910974646657766852397326, 4.03137258369416807580917582221, 5.31501265359363504375946607783, 6.16827925347434225808974143010, 6.80238051486425567189278622551, 7.57719904095778612843745925885, 8.512280805834303309378364583846, 9.557920937791690140886301999819, 10.65429995339887270788488095082

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