Properties

Label 2-980-980.879-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9094926789\)
\(L(\frac12)\) \(\approx\) \(0.9094926789\)
\(L(1)\) 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

−10.31211639276166441173760172547, −9.734317595906468772761855897325, −9.059124116061184487108592940661, −8.376424164096780162389955836766, −7.26408386523432013986932352268, −6.00489098393875909495864211230, −5.56133147093104563124879788109, −4.73177254426291277956017210099, −3.77503124429605556725499817850, −2.23193911291551178461193038426, 1.15040363770417922607503913459, 1.74059881538931158605859433554, 2.89789893294290507181932985923, 4.47006032678799768929159287214, 5.43021739980087364515215940829, 6.54948151342709462068481617692, 7.49517461690339490427166336834, 8.118157232775885370668794348112, 8.900774716059262998736264634494, 9.793968427449279027090321180298

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