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Properties

Label 2-980-140.27-c0-0-3
Degree $2$
Conductor $980$
Sign $-0.00367 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.00367 - 0.999i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.00367 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.568293807\)
\(L(\frac12)\) \(\approx\) \(1.568293807\)
\(L(1)\) 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

−10.22938687116831428977713551442, −9.743320675438621408806380572228, −8.491302795313537051447011253845, −7.64633093724011376462585373076, −7.10450258373541890793215596598, −5.89121608708959695267026830840, −5.36916412845298523616950808288, −4.54969189325605177466008925693, −3.02355130009732739476467168767, −2.34244542204240620643348345302, 1.42621620033759736223233005094, 2.48210900770174366785475672495, 3.69616442725156801111694325117, 4.69776394705816866983927498743, 5.50754086706156739360600692153, 6.41770634390120764217219026595, 7.12034478696536232940086700381, 8.783550625731888766208041630935, 9.384287478121638477805074203703, 9.964983125527364656846367174441

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