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Properties

Label 2-980-140.107-c1-0-19
Degree $2$
Conductor $980$
Sign $0.917 - 0.397i$
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  + (0.433 − 1.34i)2-s + (0.0504 − 0.188i)3-s + (−1.62 − 1.16i)4-s + (−2.13 − 0.657i)5-s + (−0.231 − 0.149i)6-s + (−2.27 + 1.68i)8-s + (2.56 + 1.48i)9-s + (−1.81 + 2.59i)10-s + (−2.70 + 1.56i)11-s + (−0.301 + 0.247i)12-s + (−2.50 + 2.50i)13-s + (−0.231 + 0.369i)15-s + (1.27 + 3.79i)16-s + (−1.04 + 3.90i)17-s + (3.10 − 2.81i)18-s + (1.64 − 2.85i)19-s + ⋯
L(s)  = 1  + (0.306 − 0.951i)2-s + (0.0291 − 0.108i)3-s + (−0.812 − 0.583i)4-s + (−0.955 − 0.294i)5-s + (−0.0945 − 0.0610i)6-s + (−0.804 + 0.594i)8-s + (0.855 + 0.493i)9-s + (−0.572 + 0.819i)10-s + (−0.816 + 0.471i)11-s + (−0.0871 + 0.0713i)12-s + (−0.694 + 0.694i)13-s + (−0.0598 + 0.0953i)15-s + (0.319 + 0.947i)16-s + (−0.253 + 0.947i)17-s + (0.731 − 0.662i)18-s + (0.377 − 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.917 - 0.397i$
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: \((2,\ 980,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866553 + 0.179870i\)
\(L(\frac12)\) \(\approx\) \(0.866553 + 0.179870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 + 1.34i)T \)
5 \( 1 + (2.13 + 0.657i)T \)
7 \( 1 \)
good3 \( 1 + (-0.0504 + 0.188i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.50 - 2.50i)T - 13iT^{2} \)
17 \( 1 + (1.04 - 3.90i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.64 + 2.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.79 + 2.35i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.55iT - 29T^{2} \)
31 \( 1 + (7.50 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.97 - 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 + (-1.17 - 1.17i)T + 43iT^{2} \)
47 \( 1 + (0.218 + 0.817i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.53 - 0.411i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.20 - 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 - 5.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.33 + 2.50i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.15iT - 71T^{2} \)
73 \( 1 + (-8.20 - 2.19i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0767 - 0.132i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.57 + 8.57i)T + 83iT^{2} \)
89 \( 1 + (-15.0 - 8.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.28 + 2.28i)T + 97iT^{2} \)
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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

−10.41902598361385923353139627028, −9.213133733055296978994162495425, −8.651558529698727478704332070700, −7.48358047587749725874985784630, −6.88861350022825874113440201467, −5.13243664891603711092134729964, −4.76889593108549259951250475556, −3.77743606313882793900493167169, −2.62460431022279543224453991591, −1.38876911098529170970798600457, 0.39678471292488307137028425346, 2.97854686242539011758096688830, 3.72771471156055103028760883151, 4.82805704794092072328010526936, 5.50082586999510596491651317765, 6.80484118752495214700928402449, 7.40874792608976858876576312240, 7.930746090924513187983228959696, 9.004016562700556962635324380969, 9.747206550651042858845045490945

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