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Properties

Label 2-980-140.83-c0-0-2
Degree $2$
Conductor $980$
Sign $-0.312 + 0.949i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.707 − 0.707i)8-s − i·9-s + (−0.382 + 0.923i)10-s + (1.30 − 1.30i)13-s − 1.00·16-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)18-s + (0.382 + 0.923i)20-s + (0.707 − 0.707i)25-s − 1.84i·26-s + 1.41i·29-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.707 − 0.707i)8-s − i·9-s + (−0.382 + 0.923i)10-s + (1.30 − 1.30i)13-s − 1.00·16-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)18-s + (0.382 + 0.923i)20-s + (0.707 − 0.707i)25-s − 1.84i·26-s + 1.41i·29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.312 + 0.949i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212577912\)
\(L(\frac12)\) \(\approx\) \(1.212577912\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 \)
good3 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 0.765iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.84iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
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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

−10.31931526721876303228458844930, −9.160282857966283140148503510869, −8.462805920839058233381433610839, −7.24869806802500116947537904596, −6.41534864090376985385323385142, −5.55727464839791360916740206510, −4.38687661902363275688993832946, −3.50833687590823589561872300268, −2.90578519949969390264162730425, −1.00331642234916133398147669158, 2.11501310886532000986495153431, 3.69875976717144904328678153472, 4.25775972358968572224867657407, 5.15403725463958803765858571904, 6.23309887393825543617919649006, 7.00734310380392975892181054277, 8.000146913959594238678010691674, 8.446476905666379842775295374084, 9.285924027332344945822100256406, 10.78193853657601427176105470986

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