Properties

Label 2-980-140.107-c1-0-2
Degree $2$
Conductor $980$
Sign $0.966 + 0.255i$
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  + (0.157 + 1.40i)2-s + (−0.737 + 2.75i)3-s + (−1.95 + 0.443i)4-s + (−2.22 + 0.187i)5-s + (−3.98 − 0.602i)6-s + (−0.931 − 2.67i)8-s + (−4.43 − 2.55i)9-s + (−0.615 − 3.10i)10-s + (−1.99 + 1.15i)11-s + (0.217 − 5.69i)12-s + (−2.36 + 2.36i)13-s + (1.12 − 6.27i)15-s + (3.60 − 1.73i)16-s + (−2.07 + 7.72i)17-s + (2.89 − 6.63i)18-s + (2.72 − 4.71i)19-s + ⋯
L(s)  = 1  + (0.111 + 0.993i)2-s + (−0.425 + 1.58i)3-s + (−0.975 + 0.221i)4-s + (−0.996 + 0.0840i)5-s + (−1.62 − 0.245i)6-s + (−0.329 − 0.944i)8-s + (−1.47 − 0.853i)9-s + (−0.194 − 0.980i)10-s + (−0.602 + 0.348i)11-s + (0.0627 − 1.64i)12-s + (−0.654 + 0.654i)13-s + (0.290 − 1.61i)15-s + (0.901 − 0.432i)16-s + (−0.502 + 1.87i)17-s + (0.682 − 1.56i)18-s + (0.624 − 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.966 + 0.255i$
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: \((2,\ 980,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133752 - 0.0173839i\)
\(L(\frac12)\) \(\approx\) \(0.133752 - 0.0173839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.157 - 1.40i)T \)
5 \( 1 + (2.22 - 0.187i)T \)
7 \( 1 \)
good3 \( 1 + (0.737 - 2.75i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.99 - 1.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.36 - 2.36i)T - 13iT^{2} \)
17 \( 1 + (2.07 - 7.72i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.72 + 4.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 0.837i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.56iT - 29T^{2} \)
31 \( 1 + (-2.17 + 1.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 - 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.11T + 41T^{2} \)
43 \( 1 + (4.62 + 4.62i)T + 43iT^{2} \)
47 \( 1 + (-2.94 - 10.9i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.533 - 0.143i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.54 + 7.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 - 3.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.75 - 1.27i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.02iT - 71T^{2} \)
73 \( 1 + (7.97 + 2.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.38 + 4.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.04 + 5.04i)T + 83iT^{2} \)
89 \( 1 + (-5.38 - 3.10i)T + (44.5 + 77.0i)T^{2} \)
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

−10.64549879598643589759424731539, −9.895935657832559315631291427758, −9.063958917583032900164366414846, −8.385438352269592934540242303047, −7.41426611792000331911544058619, −6.51759885489500176667639387968, −5.43634961286094732359166980226, −4.55967104213128033938012967564, −4.21760825319961604575185426980, −3.11639137647019123973653333131, 0.07651903297140695035110344187, 1.08419831070093835490027526754, 2.53585394688846860820327287805, 3.33738438953256606958488491122, 4.91676662573615150977339787449, 5.47631035342296724831536184566, 6.87086103264206201958290562954, 7.58838568263243391859473538951, 8.202452622972543107886560774150, 9.159850198833356394345425858384

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