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Properties

Label 2-980-35.19-c0-0-1
Degree 22
Conductor 980980
Sign 0.0633+0.997i0.0633 + 0.997i
Analytic cond. 0.4890830.489083
Root an. cond. 0.6993450.699345
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(980s/2 ΓC(s) L(s)=((0.0633+0.997i) Λ‾(1−s)\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}
Λ(s)=(980s/2 ΓC(s) L(s)=((0.0633+0.997i) Λ‾(1−s)\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: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.0633+0.997i0.0633 + 0.997i
Analytic conductor: 0.4890830.489083
Root analytic conductor: 0.6993450.699345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ980(509,⋅)\chi_{980} (509, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :0), 0.0633+0.997i)(2,\ 980,\ (\ :0),\ 0.0633 + 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97299465140.9729946514
L(12)L(\frac12) ≈\approx 0.97299465140.9729946514
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1 1
good3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T+T2 1 - T + T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−2T+T2 1 - 2T + T^{2}
73 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+2T+T2 1 + 2T + T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T+T2 1 - T + T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line