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Properties

Label 2-153-153.11-c1-0-15
Degree 22
Conductor 153153
Sign −0.437−0.899i-0.437 - 0.899i
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  + (−1.34 − 1.75i)2-s + (−0.353 − 1.69i)3-s + (−0.754 + 2.81i)4-s + (−1.95 − 0.127i)5-s + (−2.50 + 2.90i)6-s + (−0.0117 − 0.179i)7-s + (1.87 − 0.776i)8-s + (−2.75 + 1.19i)9-s + (2.40 + 3.60i)10-s + (−1.93 − 1.69i)11-s + (5.04 + 0.284i)12-s + (1.09 + 0.292i)13-s + (−0.299 + 0.263i)14-s + (0.472 + 3.35i)15-s + (1.15 + 0.665i)16-s + (4.05 + 0.759i)17-s + ⋯
L(s)  = 1  + (−0.954 − 1.24i)2-s + (−0.203 − 0.978i)3-s + (−0.377 + 1.40i)4-s + (−0.872 − 0.0572i)5-s + (−1.02 + 1.18i)6-s + (−0.00444 − 0.0678i)7-s + (0.662 − 0.274i)8-s + (−0.916 + 0.399i)9-s + (0.761 + 1.14i)10-s + (−0.583 − 0.511i)11-s + (1.45 + 0.0822i)12-s + (0.303 + 0.0812i)13-s + (−0.0801 + 0.0702i)14-s + (0.121 + 0.866i)15-s + (0.288 + 0.166i)16-s + (0.982 + 0.184i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((−0.437−0.899i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((−0.437−0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −0.437−0.899i-0.437 - 0.899i
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), −0.437−0.899i)(2,\ 153,\ (\ :1/2),\ -0.437 - 0.899i)

Particular Values

L(1)L(1) ≈\approx 0.148564+0.237377i0.148564 + 0.237377i
L(12)L(\frac12) ≈\approx 0.148564+0.237377i0.148564 + 0.237377i
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+(0.353+1.69i)T 1 + (0.353 + 1.69i)T
17 1+(−4.05−0.759i)T 1 + (-4.05 - 0.759i)T
good2 1+(1.34+1.75i)T+(−0.517+1.93i)T2 1 + (1.34 + 1.75i)T + (-0.517 + 1.93i)T^{2}
5 1+(1.95+0.127i)T+(4.95+0.652i)T2 1 + (1.95 + 0.127i)T + (4.95 + 0.652i)T^{2}
7 1+(0.0117+0.179i)T+(−6.94+0.913i)T2 1 + (0.0117 + 0.179i)T + (-6.94 + 0.913i)T^{2}
11 1+(1.93+1.69i)T+(1.43+10.9i)T2 1 + (1.93 + 1.69i)T + (1.43 + 10.9i)T^{2}
13 1+(−1.09−0.292i)T+(11.2+6.5i)T2 1 + (-1.09 - 0.292i)T + (11.2 + 6.5i)T^{2}
19 1+(5.72+2.37i)T+(13.4+13.4i)T2 1 + (5.72 + 2.37i)T + (13.4 + 13.4i)T^{2}
23 1+(1.13+3.34i)T+(−18.2+14.0i)T2 1 + (1.13 + 3.34i)T + (-18.2 + 14.0i)T^{2}
29 1+(5.61−2.76i)T+(17.6−23.0i)T2 1 + (5.61 - 2.76i)T + (17.6 - 23.0i)T^{2}
31 1+(3.35+3.82i)T+(−4.04+30.7i)T2 1 + (3.35 + 3.82i)T + (-4.04 + 30.7i)T^{2}
37 1+(1.71+8.61i)T+(−34.1+14.1i)T2 1 + (1.71 + 8.61i)T + (-34.1 + 14.1i)T^{2}
41 1+(−3.19+6.46i)T+(−24.9−32.5i)T2 1 + (-3.19 + 6.46i)T + (-24.9 - 32.5i)T^{2}
43 1+(−0.244+1.85i)T+(−41.5−11.1i)T2 1 + (-0.244 + 1.85i)T + (-41.5 - 11.1i)T^{2}
47 1+(−0.820+0.219i)T+(40.7−23.5i)T2 1 + (-0.820 + 0.219i)T + (40.7 - 23.5i)T^{2}
53 1+(4.44−10.7i)T+(−37.4−37.4i)T2 1 + (4.44 - 10.7i)T + (-37.4 - 37.4i)T^{2}
59 1+(2.50+1.92i)T+(15.2+56.9i)T2 1 + (2.50 + 1.92i)T + (15.2 + 56.9i)T^{2}
61 1+(10.8−0.709i)T+(60.4−7.96i)T2 1 + (10.8 - 0.709i)T + (60.4 - 7.96i)T^{2}
67 1+(−9.11+5.26i)T+(33.5−58.0i)T2 1 + (-9.11 + 5.26i)T + (33.5 - 58.0i)T^{2}
71 1+(−12.6+2.51i)T+(65.5−27.1i)T2 1 + (-12.6 + 2.51i)T + (65.5 - 27.1i)T^{2}
73 1+(−8.59−5.74i)T+(27.9+67.4i)T2 1 + (-8.59 - 5.74i)T + (27.9 + 67.4i)T^{2}
79 1+(−4.22+4.82i)T+(−10.3−78.3i)T2 1 + (-4.22 + 4.82i)T + (-10.3 - 78.3i)T^{2}
83 1+(−7.40+5.68i)T+(21.4−80.1i)T2 1 + (-7.40 + 5.68i)T + (21.4 - 80.1i)T^{2}
89 1+(10.2−10.2i)T−89iT2 1 + (10.2 - 10.2i)T - 89iT^{2}
97 1+(1.17+2.37i)T+(−59.0+76.9i)T2 1 + (1.17 + 2.37i)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

−12.32271664997283329497354889248, −11.01557449238666224186299522608, −10.81362662144692132045243047348, −9.132729435434640828648611880199, −8.226655173205027646472494920826, −7.46911787857747848015108356488, −5.83269820311371797663758501777, −3.70776582894326297950218436894, −2.21305785569613645413255873831, −0.36803032889545157305060827004, 3.66112271677350247940394805076, 5.15667097113071330960319845643, 6.26745567806657323773397741078, 7.66792785354940085359802397190, 8.305305556345178947786238128980, 9.481966018604418523969596106372, 10.25875488606291511991444662770, 11.36699370339894900485220258121, 12.51930800740770433490791297343, 14.23149431711959693212905696503

Graph of the ZZ-function along the critical line