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Properties

Label 2-980-140.107-c1-0-103
Degree $2$
Conductor $980$
Sign $0.199 + 0.979i$
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  + (1.41 − 0.0450i)2-s + (0.543 − 2.02i)3-s + (1.99 − 0.127i)4-s + (1.62 − 1.53i)5-s + (0.676 − 2.89i)6-s + (2.81 − 0.270i)8-s + (−1.22 − 0.704i)9-s + (2.22 − 2.24i)10-s + (0.366 − 0.211i)11-s + (0.826 − 4.11i)12-s + (1.56 − 1.56i)13-s + (−2.23 − 4.12i)15-s + (3.96 − 0.508i)16-s + (−1.18 + 4.41i)17-s + (−1.75 − 0.940i)18-s + (−3.66 + 6.35i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0318i)2-s + (0.313 − 1.17i)3-s + (0.997 − 0.0637i)4-s + (0.726 − 0.687i)5-s + (0.276 − 1.18i)6-s + (0.995 − 0.0955i)8-s + (−0.406 − 0.234i)9-s + (0.704 − 0.710i)10-s + (0.110 − 0.0637i)11-s + (0.238 − 1.18i)12-s + (0.433 − 0.433i)13-s + (−0.576 − 1.06i)15-s + (0.991 − 0.127i)16-s + (−0.286 + 1.07i)17-s + (−0.414 − 0.221i)18-s + (−0.841 + 1.45i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.199 + 0.979i$
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: \((2,\ 980,\ (\ :1/2),\ 0.199 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96973 - 2.42529i\)
\(L(\frac12)\) \(\approx\) \(2.96973 - 2.42529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0450i)T \)
5 \( 1 + (-1.62 + 1.53i)T \)
7 \( 1 \)
good3 \( 1 + (-0.543 + 2.02i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.366 + 0.211i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 1.56i)T - 13iT^{2} \)
17 \( 1 + (1.18 - 4.41i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.66 - 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.44 - 1.72i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.72iT - 29T^{2} \)
31 \( 1 + (1.70 - 0.982i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.20 - 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.01T + 41T^{2} \)
43 \( 1 + (-1.88 - 1.88i)T + 43iT^{2} \)
47 \( 1 + (1.70 + 6.35i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.780 - 0.209i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.35 + 2.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.925 - 1.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.70 + 2.33i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.16iT - 71T^{2} \)
73 \( 1 + (6.67 + 1.78i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.77 + 3.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.71 + 4.71i)T + 83iT^{2} \)
89 \( 1 + (10.0 + 5.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.64 - 4.64i)T + 97iT^{2} \)
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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

−10.10210627557591324793033803731, −8.612151499456453899088915274644, −8.110743617261166259274524838199, −7.13475762129148319703924726549, −6.09041994209658496516744356437, −5.83552703632970072161890511005, −4.49881140020246617614969478999, −3.47488880178215731722856980416, −1.99164128253838318982509817420, −1.53779328187008269421832677118, 2.13842501765095506263105565388, 3.00172927555134168989626595507, 4.08010430728730416557004566929, 4.69552048491651115454094337655, 5.77485511665573243642172933012, 6.59216061858816843329733108475, 7.35991776426804807060540902859, 8.754312592185264536932201957278, 9.523779646905090210566933144459, 10.29210357585925521854471254371

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