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Properties

Label 2-980-140.59-c1-0-38
Degree $2$
Conductor $980$
Sign $-0.997 - 0.0633i$
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  + (−0.707 − 1.22i)2-s + (−2.73 − 1.58i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + 4.47i·6-s + 2.82·8-s + (3.5 + 6.06i)9-s + (2.73 + 1.58i)10-s + (5.47 − 3.16i)12-s + 7.07·15-s + (−2.00 − 3.46i)16-s + (4.94 − 8.57i)18-s − 4.47i·20-s + (0.707 + 1.22i)23-s + (−7.74 − 4.47i)24-s + (2.5 − 4.33i)25-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−1.58 − 0.912i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.499i)5-s + 1.82i·6-s + 0.999·8-s + (1.16 + 2.02i)9-s + (0.866 + 0.499i)10-s + (1.58 − 0.912i)12-s + 1.82·15-s + (−0.500 − 0.866i)16-s + (1.16 − 2.02i)18-s − 0.999i·20-s + (0.147 + 0.255i)23-s + (−1.58 − 0.912i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.997 - 0.0633i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00753872 + 0.237828i\)
\(L(\frac12)\) \(\approx\) \(0.00753872 + 0.237828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (2.73 + 1.58i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + (-8.21 + 4.74i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.6 - 6.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + (-15.4 + 8.94i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.04257138277026615537867023048, −8.638293397901377234221662902362, −7.77745623132512787821592101051, −7.11627806544814219834165429313, −6.35771372127730143008455050305, −5.11709702014835413584820326101, −4.23967735770535923272289497703, −2.89753492467038051098543232863, −1.47974768009478107089592088843, −0.22621123577249492399583140723, 0.970759973161819766323144674124, 3.76904074401895361466268613771, 4.69719590798114491271384739260, 5.15399451658012768346845962023, 6.16794052114615529266807651925, 6.86503844653039864843620031909, 7.87961576661394101453849948722, 8.842778734784477504469237622865, 9.588496075028759604936999337936, 10.48232463752536450916394294136

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