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Properties

Label 2-980-35.9-c1-0-19
Degree $2$
Conductor $980$
Sign $-0.185 + 0.982i$
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  + (2.59 − 1.5i)3-s + (0.133 − 2.23i)5-s + (3 − 5.19i)9-s + (−1.5 − 2.59i)11-s + i·13-s + (−3 − 6i)15-s + (4.33 − 2.5i)17-s + (−4 + 6.92i)19-s + (−1.73 − i)23-s + (−4.96 − 0.598i)25-s − 9i·27-s + 29-s + (1 + 1.73i)31-s + (−7.79 − 4.5i)33-s + (8.66 + 5i)37-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (0.0599 − 0.998i)5-s + (1 − 1.73i)9-s + (−0.452 − 0.783i)11-s + 0.277i·13-s + (−0.774 − 1.54i)15-s + (1.05 − 0.606i)17-s + (−0.917 + 1.58i)19-s + (−0.361 − 0.208i)23-s + (−0.992 − 0.119i)25-s − 1.73i·27-s + 0.185·29-s + (0.179 + 0.311i)31-s + (−1.35 − 0.783i)33-s + (1.42 + 0.821i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.185 + 0.982i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.185 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64746 - 1.98781i\)
\(L(\frac12)\) \(\approx\) \(1.64746 - 1.98781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (-4.33 + 2.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.66 - 5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.5 - 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3iT - 97T^{2} \)
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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

−9.551512034290190293324522526241, −8.667312171599813521900788112716, −8.158971344422432364903214444130, −7.66287278232485160802656445904, −6.45492950259355283937012833394, −5.51666822624580482865690215764, −4.18414552022219464378180601435, −3.25280095826607264231692118461, −2.13953807299700065005759460834, −1.05431700983908253886378584712, 2.21350996757624950219245770409, 2.84909332267963434753625479172, 3.84026228913242950129517929817, 4.65023441726917074162145194064, 5.94043774516656480927488681283, 7.21250933477752441130218541781, 7.75074184773112457121461601321, 8.657490878952809464549848455675, 9.445539821258872373899085040339, 10.26026453260722197668040557818

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