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Properties

Label 2-981-327.326-c0-0-1
Degree $2$
Conductor $981$
Sign $0.577 - 0.816i$
Analytic cond. $0.489582$
Root an. cond. $0.699701$
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  + 2-s + 1.41i·5-s + 7-s − 8-s + 1.41i·10-s + 1.41i·13-s + 14-s − 16-s − 17-s − 1.41i·19-s + 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯
L(s)  = 1  + 2-s + 1.41i·5-s + 7-s − 8-s + 1.41i·10-s + 1.41i·13-s + 14-s − 16-s − 17-s − 1.41i·19-s + 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(981\)    =    \(3^{2} \cdot 109\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(0.489582\)
Root analytic conductor: \(0.699701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{981} (980, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 981,\ (\ :0),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.533509793\)
\(L(\frac12)\) \(\approx\) \(1.533509793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
109 \( 1 - T \)
good2 \( 1 - T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.74663947989589951931268354125, −9.324345637051428307308548289705, −8.867304082089176527619433148591, −7.51047513547281795869911579275, −6.76086190507328051084022751702, −6.10426306347206423005389140422, −4.77122937907628909459488817351, −4.35796280912732380919952418850, −3.08468252826340446715338004736, −2.21892000096672711245653318261, 1.24759465372194726074540865491, 2.89495466716610130722555927148, 4.18552752286383439917806910603, 4.81745562616014935918832703365, 5.40934160941686382023532527703, 6.26149998094942425942717335482, 7.82217113020343557526561979391, 8.382074114355067426487825222951, 9.063264189166380558283060267607, 10.06298496255351722159907515631

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