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Properties

Label 2-700-100.87-c1-0-3
Degree 22
Conductor 700700
Sign −0.802−0.596i-0.802 - 0.596i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
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  + (0.940 − 1.05i)2-s + (−1.38 + 2.72i)3-s + (−0.229 − 1.98i)4-s + (−0.521 − 2.17i)5-s + (1.56 + 4.02i)6-s + (0.707 − 0.707i)7-s + (−2.31 − 1.62i)8-s + (−3.72 − 5.12i)9-s + (−2.78 − 1.49i)10-s + (−3.29 + 4.52i)11-s + (5.72 + 2.12i)12-s + (−0.466 + 2.94i)13-s + (−0.0814 − 1.41i)14-s + (6.64 + 1.59i)15-s + (−3.89 + 0.913i)16-s + (−0.336 + 0.171i)17-s + ⋯
L(s)  = 1  + (0.665 − 0.746i)2-s + (−0.800 + 1.57i)3-s + (−0.114 − 0.993i)4-s + (−0.233 − 0.972i)5-s + (0.640 + 1.64i)6-s + (0.267 − 0.267i)7-s + (−0.818 − 0.574i)8-s + (−1.24 − 1.70i)9-s + (−0.881 − 0.472i)10-s + (−0.992 + 1.36i)11-s + (1.65 + 0.614i)12-s + (−0.129 + 0.816i)13-s + (−0.0217 − 0.377i)14-s + (1.71 + 0.411i)15-s + (−0.973 + 0.228i)16-s + (−0.0817 + 0.0416i)17-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.802−0.596i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((−0.802−0.596i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.802−0.596i-0.802 - 0.596i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(687,⋅)\chi_{700} (687, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), −0.802−0.596i)(2,\ 700,\ (\ :1/2),\ -0.802 - 0.596i)

Particular Values

L(1)L(1) ≈\approx 0.0792863+0.239687i0.0792863 + 0.239687i
L(12)L(\frac12) ≈\approx 0.0792863+0.239687i0.0792863 + 0.239687i
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+(−0.940+1.05i)T 1 + (-0.940 + 1.05i)T
5 1+(0.521+2.17i)T 1 + (0.521 + 2.17i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(1.38−2.72i)T+(−1.76−2.42i)T2 1 + (1.38 - 2.72i)T + (-1.76 - 2.42i)T^{2}
11 1+(3.29−4.52i)T+(−3.39−10.4i)T2 1 + (3.29 - 4.52i)T + (-3.39 - 10.4i)T^{2}
13 1+(0.466−2.94i)T+(−12.3−4.01i)T2 1 + (0.466 - 2.94i)T + (-12.3 - 4.01i)T^{2}
17 1+(0.336−0.171i)T+(9.99−13.7i)T2 1 + (0.336 - 0.171i)T + (9.99 - 13.7i)T^{2}
19 1+(1.65−5.08i)T+(−15.3−11.1i)T2 1 + (1.65 - 5.08i)T + (-15.3 - 11.1i)T^{2}
23 1+(0.403+2.54i)T+(−21.8+7.10i)T2 1 + (0.403 + 2.54i)T + (-21.8 + 7.10i)T^{2}
29 1+(4.85−1.57i)T+(23.4−17.0i)T2 1 + (4.85 - 1.57i)T + (23.4 - 17.0i)T^{2}
31 1+(3.80+1.23i)T+(25.0+18.2i)T2 1 + (3.80 + 1.23i)T + (25.0 + 18.2i)T^{2}
37 1+(9.90+1.56i)T+(35.1+11.4i)T2 1 + (9.90 + 1.56i)T + (35.1 + 11.4i)T^{2}
41 1+(−7.26+5.28i)T+(12.6−38.9i)T2 1 + (-7.26 + 5.28i)T + (12.6 - 38.9i)T^{2}
43 1+(6.36+6.36i)T+43iT2 1 + (6.36 + 6.36i)T + 43iT^{2}
47 1+(−2.63−1.34i)T+(27.6+38.0i)T2 1 + (-2.63 - 1.34i)T + (27.6 + 38.0i)T^{2}
53 1+(−4.35−2.21i)T+(31.1+42.8i)T2 1 + (-4.35 - 2.21i)T + (31.1 + 42.8i)T^{2}
59 1+(−2.45+1.78i)T+(18.2−56.1i)T2 1 + (-2.45 + 1.78i)T + (18.2 - 56.1i)T^{2}
61 1+(−11.3−8.26i)T+(18.8+58.0i)T2 1 + (-11.3 - 8.26i)T + (18.8 + 58.0i)T^{2}
67 1+(2.52+4.96i)T+(−39.3+54.2i)T2 1 + (2.52 + 4.96i)T + (-39.3 + 54.2i)T^{2}
71 1+(1.25−0.407i)T+(57.4−41.7i)T2 1 + (1.25 - 0.407i)T + (57.4 - 41.7i)T^{2}
73 1+(8.73−1.38i)T+(69.4−22.5i)T2 1 + (8.73 - 1.38i)T + (69.4 - 22.5i)T^{2}
79 1+(−4.91−15.1i)T+(−63.9+46.4i)T2 1 + (-4.91 - 15.1i)T + (-63.9 + 46.4i)T^{2}
83 1+(4.71−2.40i)T+(48.7−67.1i)T2 1 + (4.71 - 2.40i)T + (48.7 - 67.1i)T^{2}
89 1+(2.60−3.58i)T+(−27.5−84.6i)T2 1 + (2.60 - 3.58i)T + (-27.5 - 84.6i)T^{2}
97 1+(−2.29+4.50i)T+(−57.0−78.4i)T2 1 + (-2.29 + 4.50i)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

−10.67904676079406391447464421340, −10.18254309304961783465496075445, −9.464127224623540853806268814874, −8.653293083104834903444650325794, −7.11639297824513531458061035580, −5.60242506057504972864171846401, −5.18827123226571703874782287549, −4.24647468709828200312522055357, −3.89341298450498206177232123366, −1.97126351926316539223384274807, 0.10892447123631394018583086052, 2.39095654950568373322288937464, 3.29585233256762810908054019052, 5.15622533767269610335104254531, 5.77550174734208546237964925304, 6.51811168827335091631832614229, 7.36869346078518977997906271286, 7.88070437173008213254290102287, 8.692744280382479299116263712013, 10.56569434630294132801713489612

Graph of the ZZ-function along the critical line