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Properties

Label 2-980-140.139-c1-0-45
Degree $2$
Conductor $980$
Sign $-0.0861 + 0.996i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0861 + 0.996i$
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: \((2,\ 980,\ (\ :1/2),\ -0.0861 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828558 - 0.903296i\)
\(L(\frac12)\) \(\approx\) \(0.828558 - 0.903296i\)
\(L(\frac{3}{2})\) 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

−10.00726568327478849280846414432, −8.876700516735536002862398497588, −8.433665386229414572695529943235, −7.36877299994774510742742996909, −6.44299976193206947277259805139, −5.02105072362379727979668675483, −4.15550958319313131659879602478, −3.56548043117442713485720273588, −2.19493410253443957997810086737, −0.76958246808834459789797654455, 1.07604508229948423558510163813, 3.25845196593030679643956339127, 4.35737656820254797662176307280, 4.74358329579208431786104306368, 6.29826363159247466147462109109, 6.74379297668068268695138895594, 7.75544664004923903713525096496, 8.370931331061145067231831130305, 9.097382778955239430294992737742, 10.11512519146471876127914649561

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