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Properties

Label 2-74-37.27-c1-0-1
Degree 22
Conductor 7474
Sign 0.783−0.621i0.783 - 0.621i
Analytic cond. 0.5908920.590892
Root an. cond. 0.7686950.768695
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  + (0.866 + 0.5i)2-s + (0.366 + 0.633i)3-s + (0.499 + 0.866i)4-s + (−1.5 + 0.866i)5-s + 0.732i·6-s + (−2 − 3.46i)7-s + 0.999i·8-s + (1.23 − 2.13i)9-s − 1.73·10-s + 4.73·11-s + (−0.366 + 0.633i)12-s + (−5.19 + 3i)13-s − 3.99i·14-s + (−1.09 − 0.633i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.211 + 0.366i)3-s + (0.249 + 0.433i)4-s + (−0.670 + 0.387i)5-s + 0.298i·6-s + (−0.755 − 1.30i)7-s + 0.353i·8-s + (0.410 − 0.711i)9-s − 0.547·10-s + 1.42·11-s + (−0.105 + 0.183i)12-s + (−1.44 + 0.832i)13-s − 1.06i·14-s + (−0.283 − 0.163i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.210i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.783−0.621i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+1/2) L(s)=((0.783−0.621i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.783−0.621i0.783 - 0.621i
Analytic conductor: 0.5908920.590892
Root analytic conductor: 0.7686950.768695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ74(27,⋅)\chi_{74} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :1/2), 0.783−0.621i)(2,\ 74,\ (\ :1/2),\ 0.783 - 0.621i)

Particular Values

L(1)L(1) ≈\approx 1.11787+0.389595i1.11787 + 0.389595i
L(12)L(\frac12) ≈\approx 1.11787+0.389595i1.11787 + 0.389595i
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
37 1+(0.5−6.06i)T 1 + (0.5 - 6.06i)T
good3 1+(−0.366−0.633i)T+(−1.5+2.59i)T2 1 + (-0.366 - 0.633i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.5−0.866i)T+(2.5−4.33i)T2 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2}
7 1+(2+3.46i)T+(−3.5+6.06i)T2 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
13 1+(5.19−3i)T+(6.5−11.2i)T2 1 + (5.19 - 3i)T + (6.5 - 11.2i)T^{2}
17 1+(1.5+0.866i)T+(8.5+14.7i)T2 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2}
19 1+(1.09−0.633i)T+(9.5−16.4i)T2 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2}
23 1+1.26iT−23T2 1 + 1.26iT - 23T^{2}
29 1−4.26iT−29T2 1 - 4.26iT - 29T^{2}
31 1−1.26iT−31T2 1 - 1.26iT - 31T^{2}
41 1+(0.232+0.401i)T+(−20.5+35.5i)T2 1 + (0.232 + 0.401i)T + (-20.5 + 35.5i)T^{2}
43 1+9.46iT−43T2 1 + 9.46iT - 43T^{2}
47 1−11.6T+47T2 1 - 11.6T + 47T^{2}
53 1+(−1.26+2.19i)T+(−26.5−45.8i)T2 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.19+1.26i)T+(29.5+51.0i)T2 1 + (2.19 + 1.26i)T + (29.5 + 51.0i)T^{2}
61 1+(−12.6+7.33i)T+(30.5−52.8i)T2 1 + (-12.6 + 7.33i)T + (30.5 - 52.8i)T^{2}
67 1+(3.09+5.36i)T+(−33.5+58.0i)T2 1 + (3.09 + 5.36i)T + (-33.5 + 58.0i)T^{2}
71 1+(1.26+2.19i)T+(−35.5+61.4i)T2 1 + (1.26 + 2.19i)T + (-35.5 + 61.4i)T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1+(7.09−4.09i)T+(39.5−68.4i)T2 1 + (7.09 - 4.09i)T + (39.5 - 68.4i)T^{2}
83 1+(5.83−10.0i)T+(−41.5−71.8i)T2 1 + (5.83 - 10.0i)T + (-41.5 - 71.8i)T^{2}
89 1+(−4.5−2.59i)T+(44.5+77.0i)T2 1 + (-4.5 - 2.59i)T + (44.5 + 77.0i)T^{2}
97 1+5.19iT−97T2 1 + 5.19iT - 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

−14.62734859744811519789407056604, −13.93607461460691835680073444993, −12.52591161735497669367802514914, −11.64267762420174043955853688278, −10.19986489249041837987230741306, −9.110551508020049280683057617745, −7.12647051850906816776564857636, −6.78100534759549792262131656320, −4.35454702863015530219870428384, −3.58944060761563235987323413755, 2.51376875735871320420041919262, 4.35134283526187355372783058123, 5.88587712832323210442287172801, 7.36174003860689635232227643703, 8.811045922484158659445534165781, 9.986894660542854734188828077346, 11.65577624120736091332227575421, 12.36378321700036164341244961116, 13.07385757784209236748937081089, 14.49399322052545502238429443521

Graph of the ZZ-function along the critical line