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Properties

Label 2-70-35.18-c2-0-1
Degree 22
Conductor 7070
Sign 0.687+0.725i0.687 + 0.725i
Analytic cond. 1.907361.90736
Root an. cond. 1.381071.38107
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−5.05 + 1.35i)3-s + (1.73 + i)4-s + (4.84 − 1.23i)5-s + 7.39·6-s + (1.26 − 6.88i)7-s + (−1.99 − 2i)8-s + (15.8 − 9.16i)9-s + (−7.07 − 0.0936i)10-s + (5.56 − 9.64i)11-s + (−10.1 − 2.70i)12-s + (9.62 + 9.62i)13-s + (−4.25 + 8.94i)14-s + (−22.8 + 12.7i)15-s + (1.99 + 3.46i)16-s + (−2.29 − 8.55i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−1.68 + 0.451i)3-s + (0.433 + 0.250i)4-s + (0.969 − 0.246i)5-s + 1.23·6-s + (0.181 − 0.983i)7-s + (−0.249 − 0.250i)8-s + (1.76 − 1.01i)9-s + (−0.707 − 0.00936i)10-s + (0.506 − 0.876i)11-s + (−0.841 − 0.225i)12-s + (0.740 + 0.740i)13-s + (−0.303 + 0.638i)14-s + (−1.52 + 0.851i)15-s + (0.124 + 0.216i)16-s + (−0.134 − 0.503i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.687+0.725i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1) L(s)=((0.687+0.725i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.687+0.725i0.687 + 0.725i
Analytic conductor: 1.907361.90736
Root analytic conductor: 1.381071.38107
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ70(53,⋅)\chi_{70} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1), 0.687+0.725i)(2,\ 70,\ (\ :1),\ 0.687 + 0.725i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.592232−0.254662i0.592232 - 0.254662i
L(12)L(\frac12) ≈\approx 0.592232−0.254662i0.592232 - 0.254662i
L(2)L(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+(1.36+0.366i)T 1 + (1.36 + 0.366i)T
5 1+(−4.84+1.23i)T 1 + (-4.84 + 1.23i)T
7 1+(−1.26+6.88i)T 1 + (-1.26 + 6.88i)T
good3 1+(5.05−1.35i)T+(7.79−4.5i)T2 1 + (5.05 - 1.35i)T + (7.79 - 4.5i)T^{2}
11 1+(−5.56+9.64i)T+(−60.5−104.i)T2 1 + (-5.56 + 9.64i)T + (-60.5 - 104. i)T^{2}
13 1+(−9.62−9.62i)T+169iT2 1 + (-9.62 - 9.62i)T + 169iT^{2}
17 1+(2.29+8.55i)T+(−250.+144.5i)T2 1 + (2.29 + 8.55i)T + (-250. + 144.5i)T^{2}
19 1+(5.79−3.34i)T+(180.5−312.i)T2 1 + (5.79 - 3.34i)T + (180.5 - 312. i)T^{2}
23 1+(−7.37+27.5i)T+(−458.−264.5i)T2 1 + (-7.37 + 27.5i)T + (-458. - 264.5i)T^{2}
29 1+29.0iT−841T2 1 + 29.0iT - 841T^{2}
31 1+(11.9−20.6i)T+(−480.5−832.i)T2 1 + (11.9 - 20.6i)T + (-480.5 - 832. i)T^{2}
37 1+(14.9+4.01i)T+(1.18e3+684.5i)T2 1 + (14.9 + 4.01i)T + (1.18e3 + 684.5i)T^{2}
41 1+9.18T+1.68e3T2 1 + 9.18T + 1.68e3T^{2}
43 1+(−55.1−55.1i)T+1.84e3iT2 1 + (-55.1 - 55.1i)T + 1.84e3iT^{2}
47 1+(−8.55−2.29i)T+(1.91e3+1.10e3i)T2 1 + (-8.55 - 2.29i)T + (1.91e3 + 1.10e3i)T^{2}
53 1+(−13.8+3.69i)T+(2.43e3−1.40e3i)T2 1 + (-13.8 + 3.69i)T + (2.43e3 - 1.40e3i)T^{2}
59 1+(−67.7−39.1i)T+(1.74e3+3.01e3i)T2 1 + (-67.7 - 39.1i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(10.7+18.5i)T+(−1.86e3+3.22e3i)T2 1 + (10.7 + 18.5i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−13.7−51.3i)T+(−3.88e3+2.24e3i)T2 1 + (-13.7 - 51.3i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1+101.T+5.04e3T2 1 + 101.T + 5.04e3T^{2}
73 1+(101.−27.1i)T+(4.61e3−2.66e3i)T2 1 + (101. - 27.1i)T + (4.61e3 - 2.66e3i)T^{2}
79 1+(11.7−6.81i)T+(3.12e3−5.40e3i)T2 1 + (11.7 - 6.81i)T + (3.12e3 - 5.40e3i)T^{2}
83 1+(−28.4−28.4i)T+6.88e3iT2 1 + (-28.4 - 28.4i)T + 6.88e3iT^{2}
89 1+(6.56−3.78i)T+(3.96e3−6.85e3i)T2 1 + (6.56 - 3.78i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−74.4+74.4i)T−9.40e3iT2 1 + (-74.4 + 74.4i)T - 9.40e3iT^{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

−14.23556457228354558843810188002, −13.01071172733829426726119933910, −11.65665442858456498714462306048, −10.87091066026654202247414355085, −10.10912447787739286602147395519, −8.878723893836721308077086107996, −6.80932824096324545775930108538, −5.94635534920245887400273621639, −4.35973074840989362381215496509, −0.982621961232194281795932307422, 1.66957993569840217454725678751, 5.35489780144568406195060000576, 6.08936180640168913673740925466, 7.19054181205584751045295252488, 9.013113693769120650060651668670, 10.28261093601380008383943834972, 11.15621467859496905041500501865, 12.22650033885301278133693879841, 13.15556948108208030269289117765, 14.90658029455703333613567735971

Graph of the ZZ-function along the critical line