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Properties

Label 2-234-117.11-c1-0-5
Degree 22
Conductor 234234
Sign 0.722+0.690i0.722 + 0.690i
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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  + (−0.707 + 0.707i)2-s + (0.0694 − 1.73i)3-s − 1.00i·4-s + (0.339 + 0.0908i)5-s + (1.17 + 1.27i)6-s + (2.97 + 0.797i)7-s + (0.707 + 0.707i)8-s + (−2.99 − 0.240i)9-s + (−0.304 + 0.175i)10-s + (−2.35 − 2.35i)11-s + (−1.73 − 0.0694i)12-s + (2.60 − 2.49i)13-s + (−2.66 + 1.54i)14-s + (0.180 − 0.580i)15-s − 1.00·16-s + (3.87 − 6.71i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0401 − 0.999i)3-s − 0.500i·4-s + (0.151 + 0.0406i)5-s + (0.479 + 0.519i)6-s + (1.12 + 0.301i)7-s + (0.250 + 0.250i)8-s + (−0.996 − 0.0801i)9-s + (−0.0961 + 0.0555i)10-s + (−0.709 − 0.709i)11-s + (−0.499 − 0.0200i)12-s + (0.723 − 0.690i)13-s + (−0.713 + 0.411i)14-s + (0.0466 − 0.149i)15-s − 0.250·16-s + (0.940 − 1.62i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.722+0.690i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.722+0.690i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.722+0.690i0.722 + 0.690i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(11,⋅)\chi_{234} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.722+0.690i)(2,\ 234,\ (\ :1/2),\ 0.722 + 0.690i)

Particular Values

L(1)L(1) ≈\approx 0.981433−0.393538i0.981433 - 0.393538i
L(12)L(\frac12) ≈\approx 0.981433−0.393538i0.981433 - 0.393538i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1+(−0.0694+1.73i)T 1 + (-0.0694 + 1.73i)T
13 1+(−2.60+2.49i)T 1 + (-2.60 + 2.49i)T
good5 1+(−0.339−0.0908i)T+(4.33+2.5i)T2 1 + (-0.339 - 0.0908i)T + (4.33 + 2.5i)T^{2}
7 1+(−2.97−0.797i)T+(6.06+3.5i)T2 1 + (-2.97 - 0.797i)T + (6.06 + 3.5i)T^{2}
11 1+(2.35+2.35i)T+11iT2 1 + (2.35 + 2.35i)T + 11iT^{2}
17 1+(−3.87+6.71i)T+(−8.5−14.7i)T2 1 + (-3.87 + 6.71i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.67+0.984i)T+(16.4−9.5i)T2 1 + (-3.67 + 0.984i)T + (16.4 - 9.5i)T^{2}
23 1+(3.34−5.78i)T+(−11.5−19.9i)T2 1 + (3.34 - 5.78i)T + (-11.5 - 19.9i)T^{2}
29 1−5.68iT−29T2 1 - 5.68iT - 29T^{2}
31 1+(−0.293+1.09i)T+(−26.8−15.5i)T2 1 + (-0.293 + 1.09i)T + (-26.8 - 15.5i)T^{2}
37 1+(−8.90−2.38i)T+(32.0+18.5i)T2 1 + (-8.90 - 2.38i)T + (32.0 + 18.5i)T^{2}
41 1+(0.678+2.53i)T+(−35.5+20.5i)T2 1 + (0.678 + 2.53i)T + (-35.5 + 20.5i)T^{2}
43 1+(3.68−2.12i)T+(21.5−37.2i)T2 1 + (3.68 - 2.12i)T + (21.5 - 37.2i)T^{2}
47 1+(4.17−1.11i)T+(40.7−23.5i)T2 1 + (4.17 - 1.11i)T + (40.7 - 23.5i)T^{2}
53 1−5.65iT−53T2 1 - 5.65iT - 53T^{2}
59 1+(−7.75−7.75i)T+59iT2 1 + (-7.75 - 7.75i)T + 59iT^{2}
61 1+(−5.01−8.69i)T+(−30.5+52.8i)T2 1 + (-5.01 - 8.69i)T + (-30.5 + 52.8i)T^{2}
67 1+(13.3−3.56i)T+(58.0−33.5i)T2 1 + (13.3 - 3.56i)T + (58.0 - 33.5i)T^{2}
71 1+(0.343+1.28i)T+(−61.4+35.5i)T2 1 + (0.343 + 1.28i)T + (-61.4 + 35.5i)T^{2}
73 1+(2.19−2.19i)T−73iT2 1 + (2.19 - 2.19i)T - 73iT^{2}
79 1+(6.22−10.7i)T+(−39.5−68.4i)T2 1 + (6.22 - 10.7i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.05+3.92i)T+(−71.8+41.5i)T2 1 + (1.05 + 3.92i)T + (-71.8 + 41.5i)T^{2}
89 1+(−1.43+5.37i)T+(−77.0−44.5i)T2 1 + (-1.43 + 5.37i)T + (-77.0 - 44.5i)T^{2}
97 1+(0.218−0.816i)T+(−84.0−48.5i)T2 1 + (0.218 - 0.816i)T + (-84.0 - 48.5i)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

−11.74949749556696619988535725520, −11.37421569241112146358106105944, −10.02924676429996962181384901750, −8.797401747953847486025448554195, −7.894708346886868507407386716618, −7.43179512899532234650275246483, −5.84885788169194129072004235177, −5.29919743853389043924522807686, −2.88979734604609203307264607820, −1.17567566163927208183146700977, 1.91794542094610249724133875585, 3.72040521366786854538156840335, 4.68263123457604629382979297304, 5.98956188273790638995093170146, 7.86592002634925933768310759044, 8.349143827319299241181797441990, 9.696440507944400960851338306644, 10.27522809827002439799011942449, 11.17243184754610966682383352164, 11.90753967382359902090061691030

Graph of the ZZ-function along the critical line