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Properties

Label 2-980-140.123-c1-0-12
Degree $2$
Conductor $980$
Sign $-0.402 + 0.915i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.429 + 1.34i)2-s + (0.727 + 2.71i)3-s + (−1.63 + 1.15i)4-s + (−0.178 + 2.22i)5-s + (−3.34 + 2.14i)6-s + (−2.25 − 1.70i)8-s + (−4.24 + 2.44i)9-s + (−3.08 + 0.715i)10-s + (2.75 + 1.59i)11-s + (−4.32 − 3.58i)12-s + (−2.41 − 2.41i)13-s + (−6.18 + 1.13i)15-s + (1.32 − 3.77i)16-s + (0.600 + 2.24i)17-s + (−5.12 − 4.66i)18-s + (−1.39 − 2.42i)19-s + ⋯
L(s)  = 1  + (0.303 + 0.952i)2-s + (0.419 + 1.56i)3-s + (−0.815 + 0.578i)4-s + (−0.0799 + 0.996i)5-s + (−1.36 + 0.875i)6-s + (−0.798 − 0.601i)8-s + (−1.41 + 0.816i)9-s + (−0.974 + 0.226i)10-s + (0.831 + 0.479i)11-s + (−1.24 − 1.03i)12-s + (−0.670 − 0.670i)13-s + (−1.59 + 0.293i)15-s + (0.331 − 0.943i)16-s + (0.145 + 0.543i)17-s + (−1.20 − 1.09i)18-s + (−0.320 − 0.555i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.402 + 0.915i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900438 - 1.37902i\)
\(L(\frac12)\) \(\approx\) \(0.900438 - 1.37902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.429 - 1.34i)T \)
5 \( 1 + (0.178 - 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (-0.727 - 2.71i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.75 - 1.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.41 + 2.41i)T + 13iT^{2} \)
17 \( 1 + (-0.600 - 2.24i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.39 + 2.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 1.39i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.72iT - 29T^{2} \)
31 \( 1 + (-3.01 - 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 0.623i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + (3.96 - 3.96i)T - 43iT^{2} \)
47 \( 1 + (1.63 - 6.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.6 + 3.37i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.951 - 1.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.83 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 - 1.51i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.562iT - 71T^{2} \)
73 \( 1 + (-3.23 + 0.866i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.13 - 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.38 + 4.38i)T - 83iT^{2} \)
89 \( 1 + (2.51 - 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−10.29643173975928403985571228696, −9.666446517983475015622050357664, −8.982391810749996614087064976121, −8.082228314001140850299502092514, −7.13977642091738474901569521240, −6.33055759545955101053218232893, −5.19875432909687940574654448342, −4.47066162484381502137994656144, −3.56583506066019752161791153094, −2.84692853526498157709407923641, 0.68374381608363255602783659723, 1.61988020823847531788557872448, 2.59934256897813250572015466176, 3.86104759976687172265518669660, 4.92438496673197258860430436282, 5.97520751101115825762074652949, 6.88574238012162490979589423462, 7.907828247954853065995462380400, 8.790391357410452701516733615374, 9.164245165730550484962376184486

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