Properties

Label 2-980-980.39-c0-0-0
Degree $2$
Conductor $980$
Sign $0.481 - 0.876i$
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  + (−0.988 − 0.149i)2-s + (0.123 − 0.0841i)3-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.134 + 0.0648i)6-s + (0.955 + 0.294i)7-s + (−0.900 − 0.433i)8-s + (−0.357 + 0.910i)9-s + (0.0747 − 0.997i)10-s + (0.142 − 0.0440i)12-s + (−0.900 − 0.433i)14-s + (0.0931 + 0.116i)15-s + (0.826 + 0.563i)16-s + (0.488 − 0.846i)18-s + (−0.222 + 0.974i)20-s + (0.142 − 0.0440i)21-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (0.123 − 0.0841i)3-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.134 + 0.0648i)6-s + (0.955 + 0.294i)7-s + (−0.900 − 0.433i)8-s + (−0.357 + 0.910i)9-s + (0.0747 − 0.997i)10-s + (0.142 − 0.0440i)12-s + (−0.900 − 0.433i)14-s + (0.0931 + 0.116i)15-s + (0.826 + 0.563i)16-s + (0.488 − 0.846i)18-s + (−0.222 + 0.974i)20-s + (0.142 − 0.0440i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7213519599\)
\(L(\frac12)\) \(\approx\) \(0.7213519599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 + 0.149i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (-0.955 - 0.294i)T \)
good3 \( 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-1.78 - 0.268i)T + (0.955 + 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.28242364485217067125253520467, −9.661879989576771645539526115588, −8.505089089377443651543884060801, −7.907950049201490274392969679985, −7.33568834154131008234013561661, −6.22775077298781032201449211657, −5.38777050084086605081875364970, −3.85579234042117244733015360032, −2.56263658912123895284537744722, −1.88271787752673561190279379052, 0.971870497380337700571364278086, 2.20316738035554898954803748318, 3.75855523523910320958987785217, 4.93265450947127390646766418319, 5.86134770250902083395262956498, 6.82838782857256304986763482881, 7.83218872404889209816792305855, 8.637050342351906346952729371890, 8.904595219863915046174077118002, 10.01660491664782596458974011881

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