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Properties

Label 2-1232-7.4-c1-0-13
Degree 22
Conductor 12321232
Sign −0.421−0.906i-0.421 - 0.906i
Analytic cond. 9.837569.83756
Root an. cond. 3.136493.13649
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  + (−0.296 + 0.514i)3-s + (0.933 + 1.61i)5-s + (−0.0665 + 2.64i)7-s + (1.32 + 2.29i)9-s + (−0.5 + 0.866i)11-s + 4.32·13-s − 1.10·15-s + (−0.230 + 0.398i)17-s + (0.769 + 1.33i)19-s + (−1.33 − 0.819i)21-s + (−1.73 − 2.99i)23-s + (0.757 − 1.31i)25-s − 3.35·27-s − 5.78·29-s + (−0.487 + 0.844i)31-s + ⋯
L(s)  = 1  + (−0.171 + 0.296i)3-s + (0.417 + 0.723i)5-s + (−0.0251 + 0.999i)7-s + (0.441 + 0.764i)9-s + (−0.150 + 0.261i)11-s + 1.20·13-s − 0.286·15-s + (−0.0558 + 0.0967i)17-s + (0.176 + 0.305i)19-s + (−0.292 − 0.178i)21-s + (−0.360 − 0.624i)23-s + (0.151 − 0.262i)25-s − 0.645·27-s − 1.07·29-s + (−0.0875 + 0.151i)31-s + ⋯

Functional equation

Λ(s)=(1232s/2 ΓC(s) L(s)=((−0.421−0.906i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1232s/2 ΓC(s+1/2) L(s)=((−0.421−0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12321232    =    24⋅7⋅112^{4} \cdot 7 \cdot 11
Sign: −0.421−0.906i-0.421 - 0.906i
Analytic conductor: 9.837569.83756
Root analytic conductor: 3.136493.13649
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1232(529,⋅)\chi_{1232} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1232, ( :1/2), −0.421−0.906i)(2,\ 1232,\ (\ :1/2),\ -0.421 - 0.906i)

Particular Values

L(1)L(1) ≈\approx 1.6443332601.644333260
L(12)L(\frac12) ≈\approx 1.6443332601.644333260
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
7 1+(0.0665−2.64i)T 1 + (0.0665 - 2.64i)T
11 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good3 1+(0.296−0.514i)T+(−1.5−2.59i)T2 1 + (0.296 - 0.514i)T + (-1.5 - 2.59i)T^{2}
5 1+(−0.933−1.61i)T+(−2.5+4.33i)T2 1 + (-0.933 - 1.61i)T + (-2.5 + 4.33i)T^{2}
13 1−4.32T+13T2 1 - 4.32T + 13T^{2}
17 1+(0.230−0.398i)T+(−8.5−14.7i)T2 1 + (0.230 - 0.398i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.769−1.33i)T+(−9.5+16.4i)T2 1 + (-0.769 - 1.33i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.73+2.99i)T+(−11.5+19.9i)T2 1 + (1.73 + 2.99i)T + (-11.5 + 19.9i)T^{2}
29 1+5.78T+29T2 1 + 5.78T + 29T^{2}
31 1+(0.487−0.844i)T+(−15.5−26.8i)T2 1 + (0.487 - 0.844i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.133−0.230i)T+(−18.5+32.0i)T2 1 + (-0.133 - 0.230i)T + (-18.5 + 32.0i)T^{2}
41 1−0.485T+41T2 1 - 0.485T + 41T^{2}
43 1−3.70T+43T2 1 - 3.70T + 43T^{2}
47 1+(−3.23−5.59i)T+(−23.5+40.7i)T2 1 + (-3.23 - 5.59i)T + (-23.5 + 40.7i)T^{2}
53 1+(6.21−10.7i)T+(−26.5−45.8i)T2 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.39+4.14i)T+(−29.5−51.0i)T2 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.64+8.04i)T+(−30.5+52.8i)T2 1 + (4.64 + 8.04i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.51+11.2i)T+(−33.5−58.0i)T2 1 + (-6.51 + 11.2i)T + (-33.5 - 58.0i)T^{2}
71 1−6.46T+71T2 1 - 6.46T + 71T^{2}
73 1+(3.20−5.54i)T+(−36.5−63.2i)T2 1 + (3.20 - 5.54i)T + (-36.5 - 63.2i)T^{2}
79 1+(−7.54−13.0i)T+(−39.5+68.4i)T2 1 + (-7.54 - 13.0i)T + (-39.5 + 68.4i)T^{2}
83 1+6.83T+83T2 1 + 6.83T + 83T^{2}
89 1+(1.30+2.25i)T+(−44.5+77.0i)T2 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2}
97 1+5.35T+97T2 1 + 5.35T + 97T^{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

−9.983974422878940222358961781052, −9.285544311810762095593343702624, −8.354696728623244998818858675890, −7.58037552777078380708124562420, −6.44703456262121125804385164169, −5.87883281588945876080675385882, −4.96458761714490718307784340808, −3.88210641351482324329811659325, −2.70598951697083816369289414866, −1.77674510500064850294721690132, 0.75113211551770793046289809912, 1.65792698040462913562445642436, 3.45353559249725831980869933792, 4.13554697444229815571300062099, 5.32199693336945828300206705571, 6.11062866499012988151553871287, 6.98560974135041120895610171379, 7.73363777145684639083031259965, 8.766887434445903317490602140283, 9.401690401633672717544813354239

Graph of the ZZ-function along the critical line