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Properties

Label 2-980-140.139-c1-0-66
Degree $2$
Conductor $980$
Sign $0.226 + 0.974i$
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  − 1.41i·2-s − 2.00·4-s + (2.23 − 0.158i)5-s + 2.82i·8-s + 3·9-s + (−0.224 − 3.15i)10-s − 4.01·13-s + 4.00·16-s + 4.90·17-s − 4.24i·18-s + (−4.46 + 0.317i)20-s + (4.94 − 0.707i)25-s + 5.67i·26-s + 9.89·29-s − 5.65i·32-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (0.997 − 0.0708i)5-s + 1.00i·8-s + 9-s + (−0.0708 − 0.997i)10-s − 1.11·13-s + 1.00·16-s + 1.19·17-s − 0.999i·18-s + (−0.997 + 0.0708i)20-s + (0.989 − 0.141i)25-s + 1.11i·26-s + 1.83·29-s − 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.226 + 0.974i$
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: \((2,\ 980,\ (\ :1/2),\ 0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43638 - 1.14107i\)
\(L(\frac12)\) \(\approx\) \(1.43638 - 1.14107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + (-2.23 + 0.158i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4.01T + 13T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.07iT - 37T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 7.25iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 18.6iT - 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−10.02431234144885671502755118868, −9.405929126246602088173773450069, −8.355843887562121630228923343964, −7.40202496490079918058706116787, −6.28412344136081695375262608101, −5.16481355226088513716959457058, −4.56025266238916972212059351988, −3.23192940031634471032156033311, −2.20813030549432332577024390540, −1.11177865793488242244567892667, 1.24222474089827617391807364323, 2.85913017237234035586513587731, 4.30455701381110122512836155141, 5.11113559393278825025008289885, 5.92255696881851980917523226776, 6.87925305179000431581758240612, 7.43206757452755982191144057513, 8.452240315933521957831635007920, 9.385088850295493746826603219247, 10.06438662731779831054982649145

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