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Properties

Label 2-980-140.139-c1-0-83
Degree $2$
Conductor $980$
Sign $0.877 + 0.480i$
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  + (1.36 − 0.385i)2-s + 0.423i·3-s + (1.70 − 1.04i)4-s + (1.74 − 1.39i)5-s + (0.163 + 0.576i)6-s + (1.91 − 2.08i)8-s + 2.82·9-s + (1.83 − 2.57i)10-s + 4.89i·11-s + (0.443 + 0.721i)12-s + 2.54·13-s + (0.591 + 0.738i)15-s + (1.80 − 3.57i)16-s − 5.11·17-s + (3.83 − 1.08i)18-s − 6.26·19-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + 0.244i·3-s + (0.851 − 0.524i)4-s + (0.780 − 0.625i)5-s + (0.0665 + 0.235i)6-s + (0.676 − 0.736i)8-s + 0.940·9-s + (0.580 − 0.814i)10-s + 1.47i·11-s + (0.128 + 0.208i)12-s + 0.706·13-s + (0.152 + 0.190i)15-s + (0.450 − 0.892i)16-s − 1.24·17-s + (0.904 − 0.256i)18-s − 1.43·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.877 + 0.480i$
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: \((2,\ 980,\ (\ :1/2),\ 0.877 + 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.40829 - 0.871933i\)
\(L(\frac12)\) \(\approx\) \(3.40829 - 0.871933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.385i)T \)
5 \( 1 + (-1.74 + 1.39i)T \)
7 \( 1 \)
good3 \( 1 - 0.423iT - 3T^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 + 6.26T + 19T^{2} \)
23 \( 1 - 4.63T + 23T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 - 1.47T + 31T^{2} \)
37 \( 1 + 2.35iT - 37T^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 - 5.68T + 59T^{2} \)
61 \( 1 + 3.87iT - 61T^{2} \)
67 \( 1 - 0.889T + 67T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + 7.87T + 73T^{2} \)
79 \( 1 - 4.63iT - 79T^{2} \)
83 \( 1 - 4.32iT - 83T^{2} \)
89 \( 1 - 2.18iT - 89T^{2} \)
97 \( 1 - 3.42T + 97T^{2} \)
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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

−10.08144331289907090170611278297, −9.356949741947129691416287042390, −8.361534572258830552888583528444, −6.86957465747803304222051951321, −6.62304856897954652618755288538, −5.29823426016544457168380586580, −4.59199995141487616872482815481, −3.97332543248561962126757533927, −2.31675943873516435726362454523, −1.55321676568007244344693241636, 1.66678782253802865080204071754, 2.79309014095342678315071389019, 3.80075058144729022006812736136, 4.82853127387097897905600753542, 6.02707118301786514825997378751, 6.43436844308908787576074064281, 7.17251964365699594897057207787, 8.330484085943272184587698233461, 9.072158111781253152280009490405, 10.50357826956675770466078578168

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