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Properties

Label 2-110-55.13-c1-0-0
Degree 22
Conductor 110110
Sign 0.516−0.856i0.516 - 0.856i
Analytic cond. 0.8783540.878354
Root an. cond. 0.9372050.937205
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  + (−0.891 + 0.453i)2-s + (−2.20 − 0.348i)3-s + (0.587 − 0.809i)4-s + (2.18 + 0.463i)5-s + (2.11 − 0.688i)6-s + (0.620 + 3.91i)7-s + (−0.156 + 0.987i)8-s + (1.87 + 0.608i)9-s + (−2.15 + 0.579i)10-s + (2.83 + 1.71i)11-s + (−1.57 + 1.57i)12-s + (1.35 + 2.65i)13-s + (−2.33 − 3.20i)14-s + (−4.65 − 1.78i)15-s + (−0.309 − 0.951i)16-s + (2.11 − 4.15i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (−1.27 − 0.201i)3-s + (0.293 − 0.404i)4-s + (0.978 + 0.207i)5-s + (0.865 − 0.281i)6-s + (0.234 + 1.48i)7-s + (−0.0553 + 0.349i)8-s + (0.623 + 0.202i)9-s + (−0.682 + 0.183i)10-s + (0.856 + 0.516i)11-s + (−0.454 + 0.454i)12-s + (0.375 + 0.736i)13-s + (−0.623 − 0.857i)14-s + (−1.20 − 0.460i)15-s + (−0.0772 − 0.237i)16-s + (0.513 − 1.00i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.516−0.856i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((0.516−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.516−0.856i0.516 - 0.856i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(13,⋅)\chi_{110} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), 0.516−0.856i)(2,\ 110,\ (\ :1/2),\ 0.516 - 0.856i)

Particular Values

L(1)L(1) ≈\approx 0.551932+0.311841i0.551932 + 0.311841i
L(12)L(\frac12) ≈\approx 0.551932+0.311841i0.551932 + 0.311841i
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+(0.891−0.453i)T 1 + (0.891 - 0.453i)T
5 1+(−2.18−0.463i)T 1 + (-2.18 - 0.463i)T
11 1+(−2.83−1.71i)T 1 + (-2.83 - 1.71i)T
good3 1+(2.20+0.348i)T+(2.85+0.927i)T2 1 + (2.20 + 0.348i)T + (2.85 + 0.927i)T^{2}
7 1+(−0.620−3.91i)T+(−6.65+2.16i)T2 1 + (-0.620 - 3.91i)T + (-6.65 + 2.16i)T^{2}
13 1+(−1.35−2.65i)T+(−7.64+10.5i)T2 1 + (-1.35 - 2.65i)T + (-7.64 + 10.5i)T^{2}
17 1+(−2.11+4.15i)T+(−9.99−13.7i)T2 1 + (-2.11 + 4.15i)T + (-9.99 - 13.7i)T^{2}
19 1+(2.59−1.88i)T+(5.87−18.0i)T2 1 + (2.59 - 1.88i)T + (5.87 - 18.0i)T^{2}
23 1+(5.14+5.14i)T+23iT2 1 + (5.14 + 5.14i)T + 23iT^{2}
29 1+(0.0660+0.0479i)T+(8.96+27.5i)T2 1 + (0.0660 + 0.0479i)T + (8.96 + 27.5i)T^{2}
31 1+(0.600−1.84i)T+(−25.0−18.2i)T2 1 + (0.600 - 1.84i)T + (-25.0 - 18.2i)T^{2}
37 1+(5.78−0.916i)T+(35.1−11.4i)T2 1 + (5.78 - 0.916i)T + (35.1 - 11.4i)T^{2}
41 1+(−1.64−2.25i)T+(−12.6+38.9i)T2 1 + (-1.64 - 2.25i)T + (-12.6 + 38.9i)T^{2}
43 1+(−2.07+2.07i)T−43iT2 1 + (-2.07 + 2.07i)T - 43iT^{2}
47 1+(−1.92+12.1i)T+(−44.6−14.5i)T2 1 + (-1.92 + 12.1i)T + (-44.6 - 14.5i)T^{2}
53 1+(3.28−1.67i)T+(31.1−42.8i)T2 1 + (3.28 - 1.67i)T + (31.1 - 42.8i)T^{2}
59 1+(−1.48+2.04i)T+(−18.2−56.1i)T2 1 + (-1.48 + 2.04i)T + (-18.2 - 56.1i)T^{2}
61 1+(−1.70+0.554i)T+(49.3−35.8i)T2 1 + (-1.70 + 0.554i)T + (49.3 - 35.8i)T^{2}
67 1+(−2.34+2.34i)T−67iT2 1 + (-2.34 + 2.34i)T - 67iT^{2}
71 1+(1.98+6.11i)T+(−57.4+41.7i)T2 1 + (1.98 + 6.11i)T + (-57.4 + 41.7i)T^{2}
73 1+(−7.72+1.22i)T+(69.4−22.5i)T2 1 + (-7.72 + 1.22i)T + (69.4 - 22.5i)T^{2}
79 1+(−3.80+11.7i)T+(−63.9−46.4i)T2 1 + (-3.80 + 11.7i)T + (-63.9 - 46.4i)T^{2}
83 1+(−13.4−6.86i)T+(48.7+67.1i)T2 1 + (-13.4 - 6.86i)T + (48.7 + 67.1i)T^{2}
89 1+5.99iT−89T2 1 + 5.99iT - 89T^{2}
97 1+(4.98+9.78i)T+(−57.0+78.4i)T2 1 + (4.98 + 9.78i)T + (-57.0 + 78.4i)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

−14.03874244951891938644957899755, −12.28966143356969482970917376882, −11.85964271329066783678000877798, −10.66740087876794407187124065825, −9.541572763454856233493077476785, −8.643131777325919250256690337212, −6.75379019814855682517407829929, −6.08403735203626021204542198379, −5.11969926623961597769338112611, −1.98032771384966181681926318828, 1.15741953883737973505121558771, 3.97709710890615481220232866398, 5.65239090232570202116099677942, 6.59974874323114408099518144172, 8.093040034741627318201772664244, 9.577836152488659702610765905719, 10.59165627167787219599264804035, 10.97331366200917866420985435627, 12.25804558415167257561531353598, 13.35426362776811411619027073425

Graph of the ZZ-function along the critical line