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Properties

Label 2-153-153.106-c1-0-7
Degree 22
Conductor 153153
Sign 0.396−0.917i0.396 - 0.917i
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  + (1.25 + 0.727i)2-s + (0.0948 + 1.72i)3-s + (0.0574 + 0.0995i)4-s + (1.43 + 0.384i)5-s + (−1.13 + 2.24i)6-s + (1.42 − 0.382i)7-s − 2.74i·8-s + (−2.98 + 0.328i)9-s + (1.52 + 1.52i)10-s + (−5.85 + 1.56i)11-s + (−0.166 + 0.108i)12-s + (0.466 + 0.807i)13-s + (2.07 + 0.556i)14-s + (−0.529 + 2.52i)15-s + (2.10 − 3.65i)16-s + (3.65 + 1.91i)17-s + ⋯
L(s)  = 1  + (0.890 + 0.514i)2-s + (0.0547 + 0.998i)3-s + (0.0287 + 0.0497i)4-s + (0.642 + 0.172i)5-s + (−0.464 + 0.917i)6-s + (0.540 − 0.144i)7-s − 0.969i·8-s + (−0.994 + 0.109i)9-s + (0.483 + 0.483i)10-s + (−1.76 + 0.473i)11-s + (−0.0481 + 0.0314i)12-s + (0.129 + 0.224i)13-s + (0.555 + 0.148i)14-s + (−0.136 + 0.650i)15-s + (0.527 − 0.912i)16-s + (0.885 + 0.463i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((0.396−0.917i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((0.396−0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: 0.396−0.917i0.396 - 0.917i
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), 0.396−0.917i)(2,\ 153,\ (\ :1/2),\ 0.396 - 0.917i)

Particular Values

L(1)L(1) ≈\approx 1.44847+0.951758i1.44847 + 0.951758i
L(12)L(\frac12) ≈\approx 1.44847+0.951758i1.44847 + 0.951758i
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+(−0.0948−1.72i)T 1 + (-0.0948 - 1.72i)T
17 1+(−3.65−1.91i)T 1 + (-3.65 - 1.91i)T
good2 1+(−1.25−0.727i)T+(1+1.73i)T2 1 + (-1.25 - 0.727i)T + (1 + 1.73i)T^{2}
5 1+(−1.43−0.384i)T+(4.33+2.5i)T2 1 + (-1.43 - 0.384i)T + (4.33 + 2.5i)T^{2}
7 1+(−1.42+0.382i)T+(6.06−3.5i)T2 1 + (-1.42 + 0.382i)T + (6.06 - 3.5i)T^{2}
11 1+(5.85−1.56i)T+(9.52−5.5i)T2 1 + (5.85 - 1.56i)T + (9.52 - 5.5i)T^{2}
13 1+(−0.466−0.807i)T+(−6.5+11.2i)T2 1 + (-0.466 - 0.807i)T + (-6.5 + 11.2i)T^{2}
19 1+6.88iT−19T2 1 + 6.88iT - 19T^{2}
23 1+(−1.42+5.30i)T+(−19.9−11.5i)T2 1 + (-1.42 + 5.30i)T + (-19.9 - 11.5i)T^{2}
29 1+(−2.23−8.33i)T+(−25.1+14.5i)T2 1 + (-2.23 - 8.33i)T + (-25.1 + 14.5i)T^{2}
31 1+(−1.47−0.394i)T+(26.8+15.5i)T2 1 + (-1.47 - 0.394i)T + (26.8 + 15.5i)T^{2}
37 1+(−0.700+0.700i)T−37iT2 1 + (-0.700 + 0.700i)T - 37iT^{2}
41 1+(0.939−3.50i)T+(−35.5−20.5i)T2 1 + (0.939 - 3.50i)T + (-35.5 - 20.5i)T^{2}
43 1+(−0.360−0.208i)T+(21.5+37.2i)T2 1 + (-0.360 - 0.208i)T + (21.5 + 37.2i)T^{2}
47 1+(3.62−6.28i)T+(−23.5−40.7i)T2 1 + (3.62 - 6.28i)T + (-23.5 - 40.7i)T^{2}
53 1−3.69iT−53T2 1 - 3.69iT - 53T^{2}
59 1+(6.12−3.53i)T+(29.5−51.0i)T2 1 + (6.12 - 3.53i)T + (29.5 - 51.0i)T^{2}
61 1+(4.93−1.32i)T+(52.8−30.5i)T2 1 + (4.93 - 1.32i)T + (52.8 - 30.5i)T^{2}
67 1+(0.134+0.232i)T+(−33.5+58.0i)T2 1 + (0.134 + 0.232i)T + (-33.5 + 58.0i)T^{2}
71 1+(−2.80+2.80i)T−71iT2 1 + (-2.80 + 2.80i)T - 71iT^{2}
73 1+(−9.68+9.68i)T−73iT2 1 + (-9.68 + 9.68i)T - 73iT^{2}
79 1+(−4.79+1.28i)T+(68.4−39.5i)T2 1 + (-4.79 + 1.28i)T + (68.4 - 39.5i)T^{2}
83 1+(0.784+0.452i)T+(41.5+71.8i)T2 1 + (0.784 + 0.452i)T + (41.5 + 71.8i)T^{2}
89 1−0.553T+89T2 1 - 0.553T + 89T^{2}
97 1+(−3.75−14.0i)T+(−84.0+48.5i)T2 1 + (-3.75 - 14.0i)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

−13.45561941586775556539967923612, −12.49314951633187076295237013455, −10.83186132536173239489928468003, −10.28826994811875251544459969656, −9.229058124702308277503239484860, −7.83951461274888822885100858300, −6.35604286585668804975245203485, −5.15371728725854065387213326906, −4.65817467792400312217032219887, −2.90517777656255147038820595834, 2.01077005271729714740373605790, 3.26947660290853450395490265364, 5.28968288010489146764955302890, 5.77600981023168729358158160212, 7.82246111445845613618882350692, 8.194900596261601985418121613226, 9.936212594869442507059835985595, 11.21597810624708285528842473984, 12.04312173128120413211119916007, 12.92423642044410480531713013464

Graph of the ZZ-function along the critical line