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Properties

Label 2-980-980.879-c0-0-1
Degree $2$
Conductor $980$
Sign $-0.999 - 0.0213i$
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  + (0.365 − 0.930i)2-s + (0.123 − 1.64i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−1.48 − 0.716i)6-s + (−0.733 − 0.680i)7-s + (−0.900 + 0.433i)8-s + (−1.71 − 0.257i)9-s + (0.826 − 0.563i)10-s + (−1.21 + 1.12i)12-s + (−0.900 + 0.433i)14-s + (1.03 − 1.29i)15-s + (0.0747 + 0.997i)16-s + (−0.865 + 1.49i)18-s + (−0.222 − 0.974i)20-s + (−1.21 + 1.12i)21-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (0.123 − 1.64i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−1.48 − 0.716i)6-s + (−0.733 − 0.680i)7-s + (−0.900 + 0.433i)8-s + (−1.71 − 0.257i)9-s + (0.826 − 0.563i)10-s + (−1.21 + 1.12i)12-s + (−0.900 + 0.433i)14-s + (1.03 − 1.29i)15-s + (0.0747 + 0.997i)16-s + (−0.865 + 1.49i)18-s + (−0.222 − 0.974i)20-s + (−1.21 + 1.12i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 - 0.0213i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.999 - 0.0213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153440739\)
\(L(\frac12)\) \(\approx\) \(1.153440739\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (-0.826 - 0.563i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
good3 \( 1 + (-0.123 + 1.64i)T + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.658 - 1.67i)T + (-0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \)
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

−9.773986750875484688731530202954, −9.256983414801909751285021394163, −7.963625570949774439441374028572, −7.17385915107205353482813671067, −6.18921621525646712612688198430, −5.85175661144068376365870511554, −4.20645363585465964467380542299, −2.92410995037884944898748705083, −2.25473108635712317896425887144, −1.02436394065117554209196516568, 2.74248928843021205642021229049, 3.72018698904898229075092646915, 4.67618863103643636575743143966, 5.45580767162355236706041343079, 5.94756201482852171565406029736, 7.10407358884181314009566502889, 8.556376311550870640703049123497, 8.965802318510472233405364156421, 9.585368792069137773143882461030, 10.21602996304923493655933020611

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