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Properties

Label 2-1224-17.15-c1-0-1
Degree 22
Conductor 12241224
Sign −0.586−0.810i-0.586 - 0.810i
Analytic cond. 9.773689.77368
Root an. cond. 3.126293.12629
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  + (0.429 − 0.177i)5-s + (−1.62 − 0.671i)7-s + (−1.79 + 4.33i)11-s − 0.542i·13-s + (2.66 + 3.14i)17-s + (−5.16 − 5.16i)19-s + (−1.25 + 3.03i)23-s + (−3.38 + 3.38i)25-s + (−1.78 + 0.740i)29-s + (0.712 + 1.71i)31-s − 0.815·35-s + (1.30 + 3.16i)37-s + (−2.06 − 0.854i)41-s + (−2.49 + 2.49i)43-s + 10.6i·47-s + ⋯
L(s)  = 1  + (0.192 − 0.0795i)5-s + (−0.612 − 0.253i)7-s + (−0.541 + 1.30i)11-s − 0.150i·13-s + (0.646 + 0.763i)17-s + (−1.18 − 1.18i)19-s + (−0.261 + 0.632i)23-s + (−0.676 + 0.676i)25-s + (−0.331 + 0.137i)29-s + (0.127 + 0.308i)31-s − 0.137·35-s + (0.215 + 0.519i)37-s + (−0.322 − 0.133i)41-s + (−0.380 + 0.380i)43-s + 1.55i·47-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((−0.586−0.810i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s+1/2) L(s)=((−0.586−0.810i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: −0.586−0.810i-0.586 - 0.810i
Analytic conductor: 9.773689.77368
Root analytic conductor: 3.126293.12629
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1224(865,⋅)\chi_{1224} (865, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :1/2), −0.586−0.810i)(2,\ 1224,\ (\ :1/2),\ -0.586 - 0.810i)

Particular Values

L(1)L(1) ≈\approx 0.73037014950.7303701495
L(12)L(\frac12) ≈\approx 0.73037014950.7303701495
L(32)L(\frac{3}{2}) 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
3 1 1
17 1+(−2.66−3.14i)T 1 + (-2.66 - 3.14i)T
good5 1+(−0.429+0.177i)T+(3.53−3.53i)T2 1 + (-0.429 + 0.177i)T + (3.53 - 3.53i)T^{2}
7 1+(1.62+0.671i)T+(4.94+4.94i)T2 1 + (1.62 + 0.671i)T + (4.94 + 4.94i)T^{2}
11 1+(1.79−4.33i)T+(−7.77−7.77i)T2 1 + (1.79 - 4.33i)T + (-7.77 - 7.77i)T^{2}
13 1+0.542iT−13T2 1 + 0.542iT - 13T^{2}
19 1+(5.16+5.16i)T+19iT2 1 + (5.16 + 5.16i)T + 19iT^{2}
23 1+(1.25−3.03i)T+(−16.2−16.2i)T2 1 + (1.25 - 3.03i)T + (-16.2 - 16.2i)T^{2}
29 1+(1.78−0.740i)T+(20.5−20.5i)T2 1 + (1.78 - 0.740i)T + (20.5 - 20.5i)T^{2}
31 1+(−0.712−1.71i)T+(−21.9+21.9i)T2 1 + (-0.712 - 1.71i)T + (-21.9 + 21.9i)T^{2}
37 1+(−1.30−3.16i)T+(−26.1+26.1i)T2 1 + (-1.30 - 3.16i)T + (-26.1 + 26.1i)T^{2}
41 1+(2.06+0.854i)T+(28.9+28.9i)T2 1 + (2.06 + 0.854i)T + (28.9 + 28.9i)T^{2}
43 1+(2.49−2.49i)T−43iT2 1 + (2.49 - 2.49i)T - 43iT^{2}
47 1−10.6iT−47T2 1 - 10.6iT - 47T^{2}
53 1+(0.634+0.634i)T+53iT2 1 + (0.634 + 0.634i)T + 53iT^{2}
59 1+(7.78−7.78i)T−59iT2 1 + (7.78 - 7.78i)T - 59iT^{2}
61 1+(−13.1−5.46i)T+(43.1+43.1i)T2 1 + (-13.1 - 5.46i)T + (43.1 + 43.1i)T^{2}
67 1−5.80T+67T2 1 - 5.80T + 67T^{2}
71 1+(2.87+6.93i)T+(−50.2+50.2i)T2 1 + (2.87 + 6.93i)T + (-50.2 + 50.2i)T^{2}
73 1+(2.69−1.11i)T+(51.6−51.6i)T2 1 + (2.69 - 1.11i)T + (51.6 - 51.6i)T^{2}
79 1+(2.62−6.34i)T+(−55.8−55.8i)T2 1 + (2.62 - 6.34i)T + (-55.8 - 55.8i)T^{2}
83 1+(7.82+7.82i)T+83iT2 1 + (7.82 + 7.82i)T + 83iT^{2}
89 1−1.76iT−89T2 1 - 1.76iT - 89T^{2}
97 1+(−0.944+0.391i)T+(68.5−68.5i)T2 1 + (-0.944 + 0.391i)T + (68.5 - 68.5i)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

−9.946695352508077507931678391767, −9.389321901238536315852596887891, −8.348995403391411149002759494983, −7.49043635273956056888315361327, −6.76500548394779932830746508292, −5.84802993947672454217524106706, −4.87316366538164713534748014432, −3.96897709026415352783582547793, −2.81317802999086942661496456404, −1.64862031927124300177314143102, 0.29334651708571257071770460862, 2.12358988679464247621796434964, 3.17370854083185211883810737504, 4.08618516147699596862881068783, 5.42798923346110158153131900290, 6.02802633777685760087493500953, 6.81005061119282871409920403328, 8.060305758680979032881262634463, 8.454339436039799345119601006741, 9.570237744007696943269978106373

Graph of the ZZ-function along the critical line