Properties

Label 2-980-140.107-c1-0-110
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: $\chi_{980} (667, \cdot )$
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

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

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