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Properties

Label 2-370-37.28-c1-0-8
Degree 22
Conductor 370370
Sign 0.940+0.339i0.940 + 0.339i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  + (0.984 − 0.173i)2-s + (0.168 − 0.956i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s − 0.971i·6-s + (2.87 + 2.41i)7-s + (0.866 − 0.5i)8-s + (1.93 + 0.703i)9-s + (−0.5 + 0.866i)10-s + (−0.877 − 1.51i)11-s + (−0.168 − 0.956i)12-s + (−0.963 − 2.64i)13-s + (3.25 + 1.87i)14-s + (0.624 + 0.744i)15-s + (0.766 − 0.642i)16-s + (−0.345 + 0.948i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.0973 − 0.552i)3-s + (0.469 − 0.171i)4-s + (−0.287 + 0.342i)5-s − 0.396i·6-s + (1.08 + 0.912i)7-s + (0.306 − 0.176i)8-s + (0.644 + 0.234i)9-s + (−0.158 + 0.273i)10-s + (−0.264 − 0.458i)11-s + (−0.0486 − 0.276i)12-s + (−0.267 − 0.733i)13-s + (0.868 + 0.501i)14-s + (0.161 + 0.192i)15-s + (0.191 − 0.160i)16-s + (−0.0837 + 0.230i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.940+0.339i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.940+0.339i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.940+0.339i0.940 + 0.339i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(361,⋅)\chi_{370} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.940+0.339i)(2,\ 370,\ (\ :1/2),\ 0.940 + 0.339i)

Particular Values

L(1)L(1) ≈\approx 2.17764−0.380869i2.17764 - 0.380869i
L(12)L(\frac12) ≈\approx 2.17764−0.380869i2.17764 - 0.380869i
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+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
5 1+(0.642−0.766i)T 1 + (0.642 - 0.766i)T
37 1+(0.584+6.05i)T 1 + (0.584 + 6.05i)T
good3 1+(−0.168+0.956i)T+(−2.81−1.02i)T2 1 + (-0.168 + 0.956i)T + (-2.81 - 1.02i)T^{2}
7 1+(−2.87−2.41i)T+(1.21+6.89i)T2 1 + (-2.87 - 2.41i)T + (1.21 + 6.89i)T^{2}
11 1+(0.877+1.51i)T+(−5.5+9.52i)T2 1 + (0.877 + 1.51i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.963+2.64i)T+(−9.95+8.35i)T2 1 + (0.963 + 2.64i)T + (-9.95 + 8.35i)T^{2}
17 1+(0.345−0.948i)T+(−13.0−10.9i)T2 1 + (0.345 - 0.948i)T + (-13.0 - 10.9i)T^{2}
19 1+(0.438+0.0772i)T+(17.8+6.49i)T2 1 + (0.438 + 0.0772i)T + (17.8 + 6.49i)T^{2}
23 1+(1.02+0.591i)T+(11.5+19.9i)T2 1 + (1.02 + 0.591i)T + (11.5 + 19.9i)T^{2}
29 1+(3.39−1.95i)T+(14.5−25.1i)T2 1 + (3.39 - 1.95i)T + (14.5 - 25.1i)T^{2}
31 1+2.93iT−31T2 1 + 2.93iT - 31T^{2}
41 1+(2.42−0.883i)T+(31.4−26.3i)T2 1 + (2.42 - 0.883i)T + (31.4 - 26.3i)T^{2}
43 1−10.8iT−43T2 1 - 10.8iT - 43T^{2}
47 1+(6.14−10.6i)T+(−23.5−40.7i)T2 1 + (6.14 - 10.6i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.78−5.69i)T+(9.20−52.1i)T2 1 + (6.78 - 5.69i)T + (9.20 - 52.1i)T^{2}
59 1+(4.45+5.31i)T+(−10.2+58.1i)T2 1 + (4.45 + 5.31i)T + (-10.2 + 58.1i)T^{2}
61 1+(4.64+12.7i)T+(−46.7+39.2i)T2 1 + (4.64 + 12.7i)T + (-46.7 + 39.2i)T^{2}
67 1+(−6.11−5.12i)T+(11.6+65.9i)T2 1 + (-6.11 - 5.12i)T + (11.6 + 65.9i)T^{2}
71 1+(−0.190+1.08i)T+(−66.7−24.2i)T2 1 + (-0.190 + 1.08i)T + (-66.7 - 24.2i)T^{2}
73 1−11.3T+73T2 1 - 11.3T + 73T^{2}
79 1+(7.08−8.44i)T+(−13.7−77.7i)T2 1 + (7.08 - 8.44i)T + (-13.7 - 77.7i)T^{2}
83 1+(5.87+2.13i)T+(63.5+53.3i)T2 1 + (5.87 + 2.13i)T + (63.5 + 53.3i)T^{2}
89 1+(1.04+1.24i)T+(−15.4+87.6i)T2 1 + (1.04 + 1.24i)T + (-15.4 + 87.6i)T^{2}
97 1+(2.18+1.26i)T+(48.5+84.0i)T2 1 + (2.18 + 1.26i)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

−11.32605657500742656357687470757, −10.87496801397189275824835741343, −9.592362533773135777607428020459, −8.133046399075437869464673794027, −7.73239512143712856159375602097, −6.45107919474587102414303994239, −5.43167383791329942541524351283, −4.44604822523622394025811858446, −2.93244358304068360591904276489, −1.75903362107312033246205611921, 1.73702934092681793827620009347, 3.69972810706674674728196066273, 4.50008391791561609543139188109, 5.13635543632072770866849474149, 6.82760414407908321578359034415, 7.51065890804745954062793454772, 8.577156346355801407829286268841, 9.800227753211635698394788905017, 10.60000487232691161408545680056, 11.54040574379455294179020896262

Graph of the ZZ-function along the critical line