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Properties

Label 2-980-140.139-c1-0-84
Degree $2$
Conductor $980$
Sign $-0.924 + 0.381i$
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  + (−0.576 − 1.29i)2-s + 2.50i·3-s + (−1.33 + 1.48i)4-s + (0.639 − 2.14i)5-s + (3.23 − 1.44i)6-s + (2.69 + 0.867i)8-s − 3.25·9-s + (−3.13 + 0.408i)10-s − 2.25i·11-s + (−3.72 − 3.34i)12-s − 5.96·13-s + (5.35 + 1.60i)15-s + (−0.430 − 3.97i)16-s + 2.00·17-s + (1.87 + 4.20i)18-s − 7.81·19-s + ⋯
L(s)  = 1  + (−0.407 − 0.913i)2-s + 1.44i·3-s + (−0.667 + 0.744i)4-s + (0.286 − 0.958i)5-s + (1.31 − 0.588i)6-s + (0.951 + 0.306i)8-s − 1.08·9-s + (−0.991 + 0.129i)10-s − 0.678i·11-s + (−1.07 − 0.964i)12-s − 1.65·13-s + (1.38 + 0.413i)15-s + (−0.107 − 0.994i)16-s + 0.486·17-s + (0.442 + 0.991i)18-s − 1.79·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.924 + 0.381i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0649546 - 0.327196i\)
\(L(\frac12)\) \(\approx\) \(0.0649546 - 0.327196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.576 + 1.29i)T \)
5 \( 1 + (-0.639 + 2.14i)T \)
7 \( 1 \)
good3 \( 1 - 2.50iT - 3T^{2} \)
11 \( 1 + 2.25iT - 11T^{2} \)
13 \( 1 + 5.96T + 13T^{2} \)
17 \( 1 - 2.00T + 17T^{2} \)
19 \( 1 + 7.81T + 19T^{2} \)
23 \( 1 - 2.99T + 23T^{2} \)
29 \( 1 + 4.87T + 29T^{2} \)
31 \( 1 - 1.49T + 31T^{2} \)
37 \( 1 + 4.78iT - 37T^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 - 9.56iT - 47T^{2} \)
53 \( 1 + 7.06iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 1.21iT - 61T^{2} \)
67 \( 1 + 1.11T + 67T^{2} \)
71 \( 1 + 8.40iT - 71T^{2} \)
73 \( 1 + 5.88T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 4.57iT - 89T^{2} \)
97 \( 1 + 4.62T + 97T^{2} \)
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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

−9.584754198303044245688150644685, −9.157421302963026057935531984299, −8.462179886903317164411041042029, −7.45675062255404347206829313670, −5.76353021205831304842404040753, −4.82462335115564839480684767926, −4.34074343866254971839446625842, −3.27918307978313669021068826438, −2.05472960792747094235758711030, −0.16772313963424912985265124684, 1.68526772990729622009536009751, 2.62495151730960501573448190463, 4.44493138813143213237126140188, 5.57923770750810818217021181556, 6.55869829931409425640953062101, 6.96358829423518498647533905204, 7.60879145320222259471484995169, 8.316689187685849304286191507887, 9.531300496666343458793194892116, 10.12638915348308947898702682423

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