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Properties

Label 2-980-35.4-c1-0-9
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: $\chi_{980} (949, \cdot )$
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

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

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