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Properties

Label 2-74-37.12-c1-0-0
Degree 22
Conductor 7474
Sign 0.880+0.473i0.880 + 0.473i
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  + (−0.766 + 0.642i)2-s + (−1.43 − 1.20i)3-s + (0.173 − 0.984i)4-s + (3.31 − 1.20i)5-s + 1.87·6-s + (0.826 − 0.300i)7-s + (0.500 + 0.866i)8-s + (0.0923 + 0.524i)9-s + (−1.76 + 3.05i)10-s + (−1.67 − 2.89i)11-s + (−1.43 + 1.20i)12-s + (−1.11 + 6.31i)13-s + (−0.439 + 0.761i)14-s + (−6.23 − 2.27i)15-s + (−0.939 − 0.342i)16-s + (−0.520 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.831 − 0.697i)3-s + (0.0868 − 0.492i)4-s + (1.48 − 0.540i)5-s + 0.767·6-s + (0.312 − 0.113i)7-s + (0.176 + 0.306i)8-s + (0.0307 + 0.174i)9-s + (−0.558 + 0.967i)10-s + (−0.504 − 0.874i)11-s + (−0.415 + 0.348i)12-s + (−0.308 + 1.75i)13-s + (−0.117 + 0.203i)14-s + (−1.61 − 0.586i)15-s + (−0.234 − 0.0855i)16-s + (−0.126 − 0.716i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.880+0.473i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+1/2) L(s)=((0.880+0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.880+0.473i0.880 + 0.473i
Analytic conductor: 0.5908920.590892
Root analytic conductor: 0.7686950.768695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ74(49,⋅)\chi_{74} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :1/2), 0.880+0.473i)(2,\ 74,\ (\ :1/2),\ 0.880 + 0.473i)

Particular Values

L(1)L(1) ≈\approx 0.678508−0.170826i0.678508 - 0.170826i
L(12)L(\frac12) ≈\approx 0.678508−0.170826i0.678508 - 0.170826i
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+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
37 1+(3.44−5.01i)T 1 + (3.44 - 5.01i)T
good3 1+(1.43+1.20i)T+(0.520+2.95i)T2 1 + (1.43 + 1.20i)T + (0.520 + 2.95i)T^{2}
5 1+(−3.31+1.20i)T+(3.83−3.21i)T2 1 + (-3.31 + 1.20i)T + (3.83 - 3.21i)T^{2}
7 1+(−0.826+0.300i)T+(5.36−4.49i)T2 1 + (-0.826 + 0.300i)T + (5.36 - 4.49i)T^{2}
11 1+(1.67+2.89i)T+(−5.5+9.52i)T2 1 + (1.67 + 2.89i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.11−6.31i)T+(−12.2−4.44i)T2 1 + (1.11 - 6.31i)T + (-12.2 - 4.44i)T^{2}
17 1+(0.520+2.95i)T+(−15.9+5.81i)T2 1 + (0.520 + 2.95i)T + (-15.9 + 5.81i)T^{2}
19 1+(−3.55−2.98i)T+(3.29+18.7i)T2 1 + (-3.55 - 2.98i)T + (3.29 + 18.7i)T^{2}
23 1+(2.91−5.05i)T+(−11.5−19.9i)T2 1 + (2.91 - 5.05i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.63−2.83i)T+(−14.5+25.1i)T2 1 + (-1.63 - 2.83i)T + (-14.5 + 25.1i)T^{2}
31 1+1.12T+31T2 1 + 1.12T + 31T^{2}
41 1+(−1.49+8.45i)T+(−38.5−14.0i)T2 1 + (-1.49 + 8.45i)T + (-38.5 - 14.0i)T^{2}
43 1−5.61T+43T2 1 - 5.61T + 43T^{2}
47 1+(−2.56+4.44i)T+(−23.5−40.7i)T2 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.252+0.0918i)T+(40.6+34.0i)T2 1 + (0.252 + 0.0918i)T + (40.6 + 34.0i)T^{2}
59 1+(7.15+2.60i)T+(45.1+37.9i)T2 1 + (7.15 + 2.60i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.369+2.09i)T+(−57.3−20.8i)T2 1 + (-0.369 + 2.09i)T + (-57.3 - 20.8i)T^{2}
67 1+(8.01−2.91i)T+(51.3−43.0i)T2 1 + (8.01 - 2.91i)T + (51.3 - 43.0i)T^{2}
71 1+(−10.0−8.45i)T+(12.3+69.9i)T2 1 + (-10.0 - 8.45i)T + (12.3 + 69.9i)T^{2}
73 1+8.57T+73T2 1 + 8.57T + 73T^{2}
79 1+(−10.8+3.96i)T+(60.5−50.7i)T2 1 + (-10.8 + 3.96i)T + (60.5 - 50.7i)T^{2}
83 1+(−0.724−4.10i)T+(−77.9+28.3i)T2 1 + (-0.724 - 4.10i)T + (-77.9 + 28.3i)T^{2}
89 1+(7.86+2.86i)T+(68.1+57.2i)T2 1 + (7.86 + 2.86i)T + (68.1 + 57.2i)T^{2}
97 1+(−8.53+14.7i)T+(−48.5−84.0i)T2 1 + (-8.53 + 14.7i)T + (-48.5 - 84.0i)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

−14.07589548316873670257593598046, −13.72130744303775566914454541995, −12.24059055514355753039499905888, −11.22108590087094980326940917592, −9.792830713763479052593183120255, −8.920002900746058253705129010598, −7.25307361417278530826443201261, −6.10008485594824768482219715413, −5.25838212421876824016455278450, −1.59013270866072073062704600361, 2.49377108056394432966295850798, 4.96917904069463441676932877451, 6.06633479311166729227181358383, 7.81205207965268755664204507915, 9.556268847832162792380504731987, 10.36678143118228997176234470288, 10.80560483297321943769679328476, 12.37130676594942482871804590783, 13.40741425438717766319729073544, 14.79219508229132085500120868295

Graph of the ZZ-function along the critical line