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Properties

Label 2-980-140.107-c1-0-23
Degree $2$
Conductor $980$
Sign $-0.335 + 0.942i$
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  + (1.16 + 0.805i)2-s + (−0.844 + 3.15i)3-s + (0.703 + 1.87i)4-s + (−1.79 + 1.33i)5-s + (−3.52 + 2.98i)6-s + (−0.689 + 2.74i)8-s + (−6.62 − 3.82i)9-s + (−3.16 + 0.108i)10-s + (1.96 − 1.13i)11-s + (−6.49 + 0.635i)12-s + (1.38 − 1.38i)13-s + (−2.69 − 6.78i)15-s + (−3.01 + 2.63i)16-s + (0.0499 − 0.186i)17-s + (−4.62 − 9.78i)18-s + (−3.45 + 5.98i)19-s + ⋯
L(s)  = 1  + (0.822 + 0.569i)2-s + (−0.487 + 1.81i)3-s + (0.351 + 0.936i)4-s + (−0.801 + 0.597i)5-s + (−1.43 + 1.21i)6-s + (−0.243 + 0.969i)8-s + (−2.20 − 1.27i)9-s + (−0.999 + 0.0344i)10-s + (0.593 − 0.342i)11-s + (−1.87 + 0.183i)12-s + (0.383 − 0.383i)13-s + (−0.696 − 1.75i)15-s + (−0.752 + 0.658i)16-s + (0.0121 − 0.0451i)17-s + (−1.08 − 2.30i)18-s + (−0.792 + 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.335 + 0.942i$
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: \((2,\ 980,\ (\ :1/2),\ -0.335 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788837 - 1.11803i\)
\(L(\frac12)\) \(\approx\) \(0.788837 - 1.11803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.805i)T \)
5 \( 1 + (1.79 - 1.33i)T \)
7 \( 1 \)
good3 \( 1 + (0.844 - 3.15i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.96 + 1.13i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \)
17 \( 1 + (-0.0499 + 0.186i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.45 - 5.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 0.866i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.33iT - 29T^{2} \)
31 \( 1 + (0.430 - 0.248i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.37 - 0.904i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.22T + 41T^{2} \)
43 \( 1 + (-2.91 - 2.91i)T + 43iT^{2} \)
47 \( 1 + (0.645 + 2.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.98 + 1.87i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.61 - 4.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.00 + 8.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.0 - 3.21i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 + (12.6 + 3.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.66 - 9.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.591 - 0.591i)T + 83iT^{2} \)
89 \( 1 + (3.45 + 1.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.09 + 1.09i)T + 97iT^{2} \)
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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

−10.89271568652215435111478970393, −9.913280686606357718534213232370, −8.754100017712594269350773447806, −8.238003098140918503671574107408, −6.91097357314655152736624443569, −6.08419169932437230634230670007, −5.28987546957495791697793049100, −4.31823852411689537292440176282, −3.69672950701913565015977433938, −3.05544235961191904706459021179, 0.51884930038301325102693795505, 1.58351993542169002500235196083, 2.66642835254416567636322621464, 4.09130570385524224797196843114, 5.03779806295810456338168091235, 6.05106600508393321495259911140, 6.80584259437799673597742951157, 7.41016801626374010498165811825, 8.490359327873354561557977617068, 9.294821454209840232524821606463

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