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Properties

Label 2-700-140.59-c1-0-40
Degree 22
Conductor 700700
Sign −0.738+0.673i-0.738 + 0.673i
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  + (−0.620 − 1.27i)2-s + (0.573 + 0.331i)3-s + (−1.23 + 1.57i)4-s + (0.0650 − 0.934i)6-s + (−2.03 + 1.68i)7-s + (2.76 + 0.585i)8-s + (−1.28 − 2.21i)9-s + (−3.12 − 1.80i)11-s + (−1.22 + 0.496i)12-s + 5.83·13-s + (3.40 + 1.54i)14-s + (−0.971 − 3.88i)16-s + (0.684 − 1.18i)17-s + (−2.02 + 3.00i)18-s + (−2.04 − 3.54i)19-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (0.331 + 0.191i)3-s + (−0.615 + 0.788i)4-s + (0.0265 − 0.381i)6-s + (−0.770 + 0.637i)7-s + (0.978 + 0.207i)8-s + (−0.426 − 0.739i)9-s + (−0.943 − 0.544i)11-s + (−0.354 + 0.143i)12-s + 1.61·13-s + (0.911 + 0.412i)14-s + (−0.242 − 0.970i)16-s + (0.165 − 0.287i)17-s + (−0.477 + 0.707i)18-s + (−0.469 − 0.813i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.738+0.673i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((−0.738+0.673i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.738+0.673i-0.738 + 0.673i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(199,⋅)\chi_{700} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), −0.738+0.673i)(2,\ 700,\ (\ :1/2),\ -0.738 + 0.673i)

Particular Values

L(1)L(1) ≈\approx 0.285729−0.737475i0.285729 - 0.737475i
L(12)L(\frac12) ≈\approx 0.285729−0.737475i0.285729 - 0.737475i
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+(0.620+1.27i)T 1 + (0.620 + 1.27i)T
5 1 1
7 1+(2.03−1.68i)T 1 + (2.03 - 1.68i)T
good3 1+(−0.573−0.331i)T+(1.5+2.59i)T2 1 + (-0.573 - 0.331i)T + (1.5 + 2.59i)T^{2}
11 1+(3.12+1.80i)T+(5.5+9.52i)T2 1 + (3.12 + 1.80i)T + (5.5 + 9.52i)T^{2}
13 1−5.83T+13T2 1 - 5.83T + 13T^{2}
17 1+(−0.684+1.18i)T+(−8.5−14.7i)T2 1 + (-0.684 + 1.18i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.04+3.54i)T+(−9.5+16.4i)T2 1 + (2.04 + 3.54i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.62+2.81i)T+(−11.5+19.9i)T2 1 + (1.62 + 2.81i)T + (-11.5 + 19.9i)T^{2}
29 1+5.19T+29T2 1 + 5.19T + 29T^{2}
31 1+(−4.43+7.67i)T+(−15.5−26.8i)T2 1 + (-4.43 + 7.67i)T + (-15.5 - 26.8i)T^{2}
37 1+(−9.34+5.39i)T+(18.5−32.0i)T2 1 + (-9.34 + 5.39i)T + (18.5 - 32.0i)T^{2}
41 1+0.832iT−41T2 1 + 0.832iT - 41T^{2}
43 1+3.10T+43T2 1 + 3.10T + 43T^{2}
47 1+(5.97−3.44i)T+(23.5−40.7i)T2 1 + (5.97 - 3.44i)T + (23.5 - 40.7i)T^{2}
53 1+(6.42+3.70i)T+(26.5+45.8i)T2 1 + (6.42 + 3.70i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.73+6.47i)T+(−29.5−51.0i)T2 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.28+0.742i)T+(30.5−52.8i)T2 1 + (-1.28 + 0.742i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.26+2.19i)T+(−33.5−58.0i)T2 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2}
71 1−3.52iT−71T2 1 - 3.52iT - 71T^{2}
73 1+(2.58−4.47i)T+(−36.5−63.2i)T2 1 + (2.58 - 4.47i)T + (-36.5 - 63.2i)T^{2}
79 1+(9.82−5.67i)T+(39.5−68.4i)T2 1 + (9.82 - 5.67i)T + (39.5 - 68.4i)T^{2}
83 1−6.49iT−83T2 1 - 6.49iT - 83T^{2}
89 1+(8.13−4.69i)T+(44.5−77.0i)T2 1 + (8.13 - 4.69i)T + (44.5 - 77.0i)T^{2}
97 1−0.343T+97T2 1 - 0.343T + 97T^{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

−9.968106652607310298697819688357, −9.344123424437625998849397204821, −8.574888188759009550894524100691, −8.011508001712259674764775494879, −6.49010927264666948406840380177, −5.65013881125378594423071659042, −4.14041216721309526865843081583, −3.22935961226930815207683154368, −2.44178935386521456764638294788, −0.47115492514589251852561660454, 1.55388527622813467851299932056, 3.29165165744332642723050566868, 4.49379686767415125941674438980, 5.68981677399078270175759229384, 6.39712973245534738703483324314, 7.46715655163999318036708260363, 8.097128222365775733343409713842, 8.785832420454092609994871072873, 9.961062148349370251789283395818, 10.42890295161667587150906133402

Graph of the ZZ-function along the critical line