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Properties

Label 2-153-153.11-c1-0-14
Degree 22
Conductor 153153
Sign 0.112+0.993i0.112 + 0.993i
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  + (0.607 + 0.791i)2-s + (−0.893 − 1.48i)3-s + (0.259 − 0.970i)4-s + (−3.76 − 0.246i)5-s + (0.631 − 1.60i)6-s + (−0.260 − 3.97i)7-s + (2.76 − 1.14i)8-s + (−1.40 + 2.65i)9-s + (−2.08 − 3.12i)10-s + (2.97 + 2.60i)11-s + (−1.67 + 0.481i)12-s + (0.149 + 0.0401i)13-s + (2.99 − 2.62i)14-s + (2.99 + 5.80i)15-s + (0.850 + 0.490i)16-s + (−1.50 − 3.83i)17-s + ⋯
L(s)  = 1  + (0.429 + 0.559i)2-s + (−0.515 − 0.856i)3-s + (0.129 − 0.485i)4-s + (−1.68 − 0.110i)5-s + (0.258 − 0.656i)6-s + (−0.0985 − 1.50i)7-s + (0.979 − 0.405i)8-s + (−0.467 + 0.883i)9-s + (−0.660 − 0.988i)10-s + (0.896 + 0.785i)11-s + (−0.482 + 0.138i)12-s + (0.0415 + 0.0111i)13-s + (0.799 − 0.701i)14-s + (0.772 + 1.49i)15-s + (0.212 + 0.122i)16-s + (−0.364 − 0.931i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((0.112+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((0.112+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: 0.112+0.993i0.112 + 0.993i
Analytic conductor: 1.221711.22171
Root analytic conductor: 1.105311.10531
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ153(11,⋅)\chi_{153} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :1/2), 0.112+0.993i)(2,\ 153,\ (\ :1/2),\ 0.112 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 0.701201−0.626254i0.701201 - 0.626254i
L(12)L(\frac12) ≈\approx 0.701201−0.626254i0.701201 - 0.626254i
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+(0.893+1.48i)T 1 + (0.893 + 1.48i)T
17 1+(1.50+3.83i)T 1 + (1.50 + 3.83i)T
good2 1+(−0.607−0.791i)T+(−0.517+1.93i)T2 1 + (-0.607 - 0.791i)T + (-0.517 + 1.93i)T^{2}
5 1+(3.76+0.246i)T+(4.95+0.652i)T2 1 + (3.76 + 0.246i)T + (4.95 + 0.652i)T^{2}
7 1+(0.260+3.97i)T+(−6.94+0.913i)T2 1 + (0.260 + 3.97i)T + (-6.94 + 0.913i)T^{2}
11 1+(−2.97−2.60i)T+(1.43+10.9i)T2 1 + (-2.97 - 2.60i)T + (1.43 + 10.9i)T^{2}
13 1+(−0.149−0.0401i)T+(11.2+6.5i)T2 1 + (-0.149 - 0.0401i)T + (11.2 + 6.5i)T^{2}
19 1+(−3.79−1.57i)T+(13.4+13.4i)T2 1 + (-3.79 - 1.57i)T + (13.4 + 13.4i)T^{2}
23 1+(−0.314−0.926i)T+(−18.2+14.0i)T2 1 + (-0.314 - 0.926i)T + (-18.2 + 14.0i)T^{2}
29 1+(−4.46+2.20i)T+(17.6−23.0i)T2 1 + (-4.46 + 2.20i)T + (17.6 - 23.0i)T^{2}
31 1+(3.23+3.69i)T+(−4.04+30.7i)T2 1 + (3.23 + 3.69i)T + (-4.04 + 30.7i)T^{2}
37 1+(0.326+1.64i)T+(−34.1+14.1i)T2 1 + (0.326 + 1.64i)T + (-34.1 + 14.1i)T^{2}
41 1+(2.15−4.36i)T+(−24.9−32.5i)T2 1 + (2.15 - 4.36i)T + (-24.9 - 32.5i)T^{2}
43 1+(0.517−3.93i)T+(−41.5−11.1i)T2 1 + (0.517 - 3.93i)T + (-41.5 - 11.1i)T^{2}
47 1+(2.97−0.797i)T+(40.7−23.5i)T2 1 + (2.97 - 0.797i)T + (40.7 - 23.5i)T^{2}
53 1+(0.152−0.367i)T+(−37.4−37.4i)T2 1 + (0.152 - 0.367i)T + (-37.4 - 37.4i)T^{2}
59 1+(−1.37−1.05i)T+(15.2+56.9i)T2 1 + (-1.37 - 1.05i)T + (15.2 + 56.9i)T^{2}
61 1+(−7.89+0.517i)T+(60.4−7.96i)T2 1 + (-7.89 + 0.517i)T + (60.4 - 7.96i)T^{2}
67 1+(−9.67+5.58i)T+(33.5−58.0i)T2 1 + (-9.67 + 5.58i)T + (33.5 - 58.0i)T^{2}
71 1+(3.55−0.707i)T+(65.5−27.1i)T2 1 + (3.55 - 0.707i)T + (65.5 - 27.1i)T^{2}
73 1+(−12.4−8.29i)T+(27.9+67.4i)T2 1 + (-12.4 - 8.29i)T + (27.9 + 67.4i)T^{2}
79 1+(7.85−8.95i)T+(−10.3−78.3i)T2 1 + (7.85 - 8.95i)T + (-10.3 - 78.3i)T^{2}
83 1+(0.349−0.268i)T+(21.4−80.1i)T2 1 + (0.349 - 0.268i)T + (21.4 - 80.1i)T^{2}
89 1+(−3.79+3.79i)T−89iT2 1 + (-3.79 + 3.79i)T - 89iT^{2}
97 1+(3.20+6.50i)T+(−59.0+76.9i)T2 1 + (3.20 + 6.50i)T + (-59.0 + 76.9i)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

−12.81260413494433339860569406509, −11.66683390682083988736359133563, −11.13639670321642902615126732411, −9.858963374196641544645156411637, −7.907764483804992549776346549278, −7.22542963322695413404245919076, −6.66589756677905899917433596037, −4.90439806201472396157386562752, −3.96939231426915795067061418005, −0.936223430061344102498045101090, 3.15336838729616860847047603912, 3.92009578084765507313463607015, 5.17735908110958273289245531372, 6.71164750748954583504191909324, 8.320828573979728703792571680996, 8.923962457630297921519195067884, 10.68950214927747782264831403070, 11.56365049635679251228265754760, 11.89297421350653040071268130441, 12.64770711698103411443594301918

Graph of the ZZ-function along the critical line