Properties

Label 2-980-35.4-c1-0-6
Degree 22
Conductor 980980
Sign 0.9820.185i0.982 - 0.185i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 0.133i)5-s + (−1.5 − 2.59i)9-s + 4i·13-s + (3.46 + 2i)17-s + (2 + 3.46i)19-s + (6.92 − 4i)23-s + (4.96 − 0.598i)25-s − 2·29-s + (4 − 6.92i)31-s + (6.92 − 4i)37-s + 6·41-s + 8i·43-s + (3.69 + 5.59i)45-s + (−6.92 + 4i)47-s + (−2 + 3.46i)59-s + ⋯
L(s)  = 1  + (−0.998 + 0.0599i)5-s + (−0.5 − 0.866i)9-s + 1.10i·13-s + (0.840 + 0.485i)17-s + (0.458 + 0.794i)19-s + (1.44 − 0.834i)23-s + (0.992 − 0.119i)25-s − 0.371·29-s + (0.718 − 1.24i)31-s + (1.13 − 0.657i)37-s + 0.937·41-s + 1.21i·43-s + (0.550 + 0.834i)45-s + (−1.01 + 0.583i)47-s + (−0.260 + 0.450i)59-s + ⋯

Functional equation

Λ(s)=(980s/2ΓC(s)L(s)=((0.9820.185i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2ΓC(s+1/2)L(s)=((0.9820.185i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    225722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.9820.185i0.982 - 0.185i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(949,)\chi_{980} (949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.9820.185i)(2,\ 980,\ (\ :1/2),\ 0.982 - 0.185i)

Particular Values

L(1)L(1) \approx 1.26846+0.118757i1.26846 + 0.118757i
L(12)L(\frac12) \approx 1.26846+0.118757i1.26846 + 0.118757i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(2.230.133i)T 1 + (2.23 - 0.133i)T
7 1 1
good3 1+(1.5+2.59i)T2 1 + (1.5 + 2.59i)T^{2}
11 1+(5.59.52i)T2 1 + (-5.5 - 9.52i)T^{2}
13 14iT13T2 1 - 4iT - 13T^{2}
17 1+(3.462i)T+(8.5+14.7i)T2 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2}
19 1+(23.46i)T+(9.5+16.4i)T2 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}
23 1+(6.92+4i)T+(11.519.9i)T2 1 + (-6.92 + 4i)T + (11.5 - 19.9i)T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+(4+6.92i)T+(15.526.8i)T2 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1+(6.92+4i)T+(18.532.0i)T2 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2}
41 16T+41T2 1 - 6T + 41T^{2}
43 18iT43T2 1 - 8iT - 43T^{2}
47 1+(6.924i)T+(23.540.7i)T2 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2}
53 1+(26.5+45.8i)T2 1 + (26.5 + 45.8i)T^{2}
59 1+(23.46i)T+(29.551.0i)T2 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(35.19i)T+(30.5+52.8i)T2 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.924i)T+(33.5+58.0i)T2 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2}
71 112T+71T2 1 - 12T + 71T^{2}
73 1+(3.462i)T+(36.5+63.2i)T2 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2}
79 1+(2+3.46i)T+(39.5+68.4i)T2 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2}
83 183T2 1 - 83T^{2}
89 1+(5+8.66i)T+(44.5+77.0i)T2 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2}
97 112iT97T2 1 - 12iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.881095382358240547002154553866, −9.199099342813155425546326455796, −8.310126054698841747483603129600, −7.58678863588756123840577605265, −6.63730428256926712623220216119, −5.84734254600871668037574197444, −4.53978395344829857762746738944, −3.77414100600725362662326140358, −2.78372263240156959442536848385, −0.962554427255944045818718986399, 0.845848652453346445710810900749, 2.78359223102205622521061339146, 3.47530894235330696219250094753, 4.96998853271478763571430683095, 5.29699790414913958053695680384, 6.77258980269466721507198350982, 7.64046123905680919581557778955, 8.128474708784298076304312557466, 9.050566794417287782413425083481, 10.03048080107623049725194204300

Graph of the ZZ-function along the critical line