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Properties

Label 2-296-37.10-c1-0-5
Degree 22
Conductor 296296
Sign 0.825+0.564i0.825 + 0.564i
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + (−0.261 − 0.453i)3-s + (1.19 + 2.07i)5-s + (−2.55 − 4.43i)7-s + (1.36 − 2.36i)9-s + 5.32·11-s + (1.22 + 2.12i)13-s + (0.626 − 1.08i)15-s + (0.5 − 0.866i)17-s + (0.435 + 0.753i)19-s + (−1.33 + 2.32i)21-s + 5.98·23-s + (−0.364 + 0.631i)25-s − 2.99·27-s − 5.07·29-s − 5.32·31-s + ⋯
L(s)  = 1  + (−0.151 − 0.261i)3-s + (0.535 + 0.927i)5-s + (−0.967 − 1.67i)7-s + (0.454 − 0.786i)9-s + 1.60·11-s + (0.340 + 0.589i)13-s + (0.161 − 0.280i)15-s + (0.121 − 0.210i)17-s + (0.0998 + 0.172i)19-s + (−0.292 + 0.506i)21-s + 1.24·23-s + (−0.0729 + 0.126i)25-s − 0.576·27-s − 0.942·29-s − 0.956·31-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=((0.825+0.564i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=((0.825+0.564i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 0.825+0.564i0.825 + 0.564i
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ296(121,⋅)\chi_{296} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 0.825+0.564i)(2,\ 296,\ (\ :1/2),\ 0.825 + 0.564i)

Particular Values

L(1)L(1) ≈\approx 1.28393−0.397174i1.28393 - 0.397174i
L(12)L(\frac12) ≈\approx 1.28393−0.397174i1.28393 - 0.397174i
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 1
37 1+(−1.34+5.93i)T 1 + (-1.34 + 5.93i)T
good3 1+(0.261+0.453i)T+(−1.5+2.59i)T2 1 + (0.261 + 0.453i)T + (-1.5 + 2.59i)T^{2}
5 1+(−1.19−2.07i)T+(−2.5+4.33i)T2 1 + (-1.19 - 2.07i)T + (-2.5 + 4.33i)T^{2}
7 1+(2.55+4.43i)T+(−3.5+6.06i)T2 1 + (2.55 + 4.43i)T + (-3.5 + 6.06i)T^{2}
11 1−5.32T+11T2 1 - 5.32T + 11T^{2}
13 1+(−1.22−2.12i)T+(−6.5+11.2i)T2 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.5+0.866i)T+(−8.5−14.7i)T2 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.435−0.753i)T+(−9.5+16.4i)T2 1 + (-0.435 - 0.753i)T + (-9.5 + 16.4i)T^{2}
23 1−5.98T+23T2 1 - 5.98T + 23T^{2}
29 1+5.07T+29T2 1 + 5.07T + 29T^{2}
31 1+5.32T+31T2 1 + 5.32T + 31T^{2}
41 1+(−3.52−6.11i)T+(−20.5+35.5i)T2 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2}
43 1+1.39T+43T2 1 + 1.39T + 43T^{2}
47 1+12.6T+47T2 1 + 12.6T + 47T^{2}
53 1+(−0.00720+0.0124i)T+(−26.5−45.8i)T2 1 + (-0.00720 + 0.0124i)T + (-26.5 - 45.8i)T^{2}
59 1+(3.09−5.36i)T+(−29.5−51.0i)T2 1 + (3.09 - 5.36i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.81−10.0i)T+(−30.5+52.8i)T2 1 + (-5.81 - 10.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.958+1.66i)T+(−33.5+58.0i)T2 1 + (0.958 + 1.66i)T + (-33.5 + 58.0i)T^{2}
71 1+(0.310+0.537i)T+(−35.5+61.4i)T2 1 + (0.310 + 0.537i)T + (-35.5 + 61.4i)T^{2}
73 1+0.635T+73T2 1 + 0.635T + 73T^{2}
79 1+(−7.73−13.3i)T+(−39.5+68.4i)T2 1 + (-7.73 - 13.3i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.691−1.19i)T+(−41.5−71.8i)T2 1 + (0.691 - 1.19i)T + (-41.5 - 71.8i)T^{2}
89 1+(−5.52+9.57i)T+(−44.5−77.0i)T2 1 + (-5.52 + 9.57i)T + (-44.5 - 77.0i)T^{2}
97 1−6.20T+97T2 1 - 6.20T + 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

−11.50953334611138617225628191114, −10.76419346751654475128739292949, −9.706747690678274311744775317117, −9.257894204505877404814869000134, −7.27886885070470357297091868810, −6.79710451068509928979611814057, −6.19891550978095729714910976260, −4.10744856201714768422115724763, −3.38980016001362032063332799294, −1.23472202783376885707929203908, 1.75431081024579326265901795392, 3.40752441033752864148978242771, 4.99252304306547224653324108930, 5.73031527758087461453608356073, 6.73681081015423066891297789610, 8.353898921049164849073860298278, 9.302762080926139891346578753743, 9.518178616625585198562516343494, 10.99250001149921161911308441503, 11.97114813899950486163318110013

Graph of the ZZ-function along the critical line