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Properties

Label 2-980-35.9-c1-0-4
Degree $2$
Conductor $980$
Sign $-0.441 - 0.897i$
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  + (1.23 + 1.86i)5-s + (−1.5 + 2.59i)9-s + 4i·13-s + (−3.46 + 2i)17-s + (2 − 3.46i)19-s + (−6.92 − 4i)23-s + (−1.96 + 4.59i)25-s − 2·29-s + (4 + 6.92i)31-s + (−6.92 − 4i)37-s + 6·41-s + 8i·43-s + (−6.69 + 0.401i)45-s + (6.92 + 4i)47-s + (−2 − 3.46i)59-s + ⋯
L(s)  = 1  + (0.550 + 0.834i)5-s + (−0.5 + 0.866i)9-s + 1.10i·13-s + (−0.840 + 0.485i)17-s + (0.458 − 0.794i)19-s + (−1.44 − 0.834i)23-s + (−0.392 + 0.919i)25-s − 0.371·29-s + (0.718 + 1.24i)31-s + (−1.13 − 0.657i)37-s + 0.937·41-s + 1.21i·43-s + (−0.998 + 0.0599i)45-s + (1.01 + 0.583i)47-s + (−0.260 − 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.441 - 0.897i$
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: \((2,\ 980,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673493 + 1.08143i\)
\(L(\frac12)\) \(\approx\) \(0.673493 + 1.08143i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.92 + 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.41590075802329140716683173023, −9.435893523906644654222695410942, −8.686546958799970887401779089743, −7.70986617927815478916254750122, −6.76584509139964317939262838027, −6.14930804088332026789838022708, −5.07194342243141494535938170370, −4.06095558429288036321659118680, −2.70595075234222469454391472969, −1.93231285190119175792442378616, 0.55861095119285999194782798143, 2.05542799489296728166079430194, 3.37635752384361742849959081639, 4.41524035427151186101550105026, 5.70928072489352256982493990613, 5.89815279966537972382874222445, 7.27448955108484871978425618953, 8.218269068034088212390689898832, 8.910306571027824520399042147475, 9.737071380720399107831814532251

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