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Properties

Label 2-153-153.106-c1-0-1
Degree 22
Conductor 153153
Sign 0.691−0.721i0.691 - 0.721i
Analytic cond. 1.221711.22171
Root an. cond. 1.105311.10531
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  + (−1.41 − 0.814i)2-s + (0.344 + 1.69i)3-s + (0.328 + 0.568i)4-s + (3.24 + 0.870i)5-s + (0.897 − 2.67i)6-s + (−4.04 + 1.08i)7-s + 2.18i·8-s + (−2.76 + 1.16i)9-s + (−3.87 − 3.87i)10-s + (3.68 − 0.988i)11-s + (−0.852 + 0.752i)12-s + (1.96 + 3.39i)13-s + (6.60 + 1.76i)14-s + (−0.359 + 5.81i)15-s + (2.44 − 4.22i)16-s + (1.76 + 3.72i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.576i)2-s + (0.198 + 0.980i)3-s + (0.164 + 0.284i)4-s + (1.45 + 0.389i)5-s + (0.366 − 1.09i)6-s + (−1.53 + 0.410i)7-s + 0.774i·8-s + (−0.921 + 0.389i)9-s + (−1.22 − 1.22i)10-s + (1.11 − 0.297i)11-s + (−0.246 + 0.217i)12-s + (0.544 + 0.942i)13-s + (1.76 + 0.472i)14-s + (−0.0928 + 1.50i)15-s + (0.610 − 1.05i)16-s + (0.427 + 0.903i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((0.691−0.721i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((0.691−0.721i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: 0.691−0.721i0.691 - 0.721i
Analytic conductor: 1.221711.22171
Root analytic conductor: 1.105311.10531
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ153(106,⋅)\chi_{153} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :1/2), 0.691−0.721i)(2,\ 153,\ (\ :1/2),\ 0.691 - 0.721i)

Particular Values

L(1)L(1) ≈\approx 0.707152+0.301756i0.707152 + 0.301756i
L(12)L(\frac12) ≈\approx 0.707152+0.301756i0.707152 + 0.301756i
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)
bad3 1+(−0.344−1.69i)T 1 + (-0.344 - 1.69i)T
17 1+(−1.76−3.72i)T 1 + (-1.76 - 3.72i)T
good2 1+(1.41+0.814i)T+(1+1.73i)T2 1 + (1.41 + 0.814i)T + (1 + 1.73i)T^{2}
5 1+(−3.24−0.870i)T+(4.33+2.5i)T2 1 + (-3.24 - 0.870i)T + (4.33 + 2.5i)T^{2}
7 1+(4.04−1.08i)T+(6.06−3.5i)T2 1 + (4.04 - 1.08i)T + (6.06 - 3.5i)T^{2}
11 1+(−3.68+0.988i)T+(9.52−5.5i)T2 1 + (-3.68 + 0.988i)T + (9.52 - 5.5i)T^{2}
13 1+(−1.96−3.39i)T+(−6.5+11.2i)T2 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2}
19 1+0.510iT−19T2 1 + 0.510iT - 19T^{2}
23 1+(−0.309+1.15i)T+(−19.9−11.5i)T2 1 + (-0.309 + 1.15i)T + (-19.9 - 11.5i)T^{2}
29 1+(0.735+2.74i)T+(−25.1+14.5i)T2 1 + (0.735 + 2.74i)T + (-25.1 + 14.5i)T^{2}
31 1+(4.60+1.23i)T+(26.8+15.5i)T2 1 + (4.60 + 1.23i)T + (26.8 + 15.5i)T^{2}
37 1+(−4.73+4.73i)T−37iT2 1 + (-4.73 + 4.73i)T - 37iT^{2}
41 1+(0.206−0.771i)T+(−35.5−20.5i)T2 1 + (0.206 - 0.771i)T + (-35.5 - 20.5i)T^{2}
43 1+(4.94+2.85i)T+(21.5+37.2i)T2 1 + (4.94 + 2.85i)T + (21.5 + 37.2i)T^{2}
47 1+(−2.00+3.47i)T+(−23.5−40.7i)T2 1 + (-2.00 + 3.47i)T + (-23.5 - 40.7i)T^{2}
53 1+7.02iT−53T2 1 + 7.02iT - 53T^{2}
59 1+(−7.35+4.24i)T+(29.5−51.0i)T2 1 + (-7.35 + 4.24i)T + (29.5 - 51.0i)T^{2}
61 1+(−1.86+0.500i)T+(52.8−30.5i)T2 1 + (-1.86 + 0.500i)T + (52.8 - 30.5i)T^{2}
67 1+(5.48+9.49i)T+(−33.5+58.0i)T2 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2}
71 1+(7.92−7.92i)T−71iT2 1 + (7.92 - 7.92i)T - 71iT^{2}
73 1+(7.02−7.02i)T−73iT2 1 + (7.02 - 7.02i)T - 73iT^{2}
79 1+(−5.60+1.50i)T+(68.4−39.5i)T2 1 + (-5.60 + 1.50i)T + (68.4 - 39.5i)T^{2}
83 1+(−11.6−6.74i)T+(41.5+71.8i)T2 1 + (-11.6 - 6.74i)T + (41.5 + 71.8i)T^{2}
89 1−12.6T+89T2 1 - 12.6T + 89T^{2}
97 1+(0.387+1.44i)T+(−84.0+48.5i)T2 1 + (0.387 + 1.44i)T + (-84.0 + 48.5i)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

−13.20055421226066412757536275706, −11.69381412227753793886441382274, −10.64439580198914524103933655584, −9.826069665785314960335209832690, −9.351144124290480596682520570331, −8.713320175706866275647551256312, −6.43909274604704833718410316704, −5.70018782217748329035846375424, −3.60712682441561410186622812063, −2.16048679704408855882641505731, 1.12582050986989145367451558244, 3.25055342204962492338826216830, 5.88405026683343091169143741869, 6.57622789840884170837078833124, 7.49013540379581862476533040567, 8.937779082502662679875420555152, 9.423418710350320962825730372138, 10.28517882571135353150327926396, 12.17875232454320307270881840066, 13.17045941414421463761328225734

Graph of the ZZ-function along the critical line