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Properties

Label 2-980-35.19-c0-0-1
Degree $2$
Conductor $980$
Sign $0.0633 + 0.997i$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)11-s + 13-s − 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s − 27-s − 29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)47-s + (−0.499 + 0.866i)51-s + 0.999·55-s + (0.5 − 0.866i)65-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)11-s + 13-s − 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s − 27-s − 29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)47-s + (−0.499 + 0.866i)51-s + 0.999·55-s + (0.5 − 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9729946514\)
\(L(\frac12)\) \(\approx\) \(0.9729946514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.750928811917439655782890720818, −9.278169488270330955037636894760, −8.341672129661015465440518538371, −7.33054238823988134626007731971, −6.57409837971363975042621195813, −5.83011220192201039323702967651, −4.86612330879583585596712118321, −3.84292403289306529142074353153, −2.10528977391486518003779400223, −1.11693489335356370797419942188, 1.86653269932265218967370641329, 3.39607395421607154023663064518, 4.05316488690105075065812475092, 5.35520324287051263023871333747, 6.05583563326882287863200091722, 6.75994113713995892889687209992, 7.978022282460619700836904918145, 8.915752765771908252737538815198, 9.727489314371706461840374461813, 10.53777644137331446862998771678

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