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Properties

Label 2-234-117.11-c1-0-8
Degree 22
Conductor 234234
Sign 0.214+0.976i0.214 + 0.976i
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 + (−0.539 − 1.64i)3-s − 1.00i·4-s + (3.04 + 0.817i)5-s + (−1.54 − 0.781i)6-s + (2.84 + 0.761i)7-s + (−0.707 − 0.707i)8-s + (−2.41 + 1.77i)9-s + (2.73 − 1.57i)10-s + (−0.616 − 0.616i)11-s + (−1.64 + 0.539i)12-s + (−3.08 − 1.86i)13-s + (2.54 − 1.47i)14-s + (−0.301 − 5.45i)15-s − 1.00·16-s + (−1.80 + 3.11i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.311 − 0.950i)3-s − 0.500i·4-s + (1.36 + 0.365i)5-s + (−0.630 − 0.319i)6-s + (1.07 + 0.287i)7-s + (−0.250 − 0.250i)8-s + (−0.805 + 0.592i)9-s + (0.864 − 0.499i)10-s + (−0.185 − 0.185i)11-s + (−0.475 + 0.155i)12-s + (−0.855 − 0.517i)13-s + (0.681 − 0.393i)14-s + (−0.0779 − 1.40i)15-s − 0.250·16-s + (−0.436 + 0.756i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.214+0.976i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.214+0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.214+0.976i0.214 + 0.976i
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), 0.214+0.976i)(2,\ 234,\ (\ :1/2),\ 0.214 + 0.976i)

Particular Values

L(1)L(1) ≈\approx 1.35979−1.09315i1.35979 - 1.09315i
L(12)L(\frac12) ≈\approx 1.35979−1.09315i1.35979 - 1.09315i
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+(0.539+1.64i)T 1 + (0.539 + 1.64i)T
13 1+(3.08+1.86i)T 1 + (3.08 + 1.86i)T
good5 1+(−3.04−0.817i)T+(4.33+2.5i)T2 1 + (-3.04 - 0.817i)T + (4.33 + 2.5i)T^{2}
7 1+(−2.84−0.761i)T+(6.06+3.5i)T2 1 + (-2.84 - 0.761i)T + (6.06 + 3.5i)T^{2}
11 1+(0.616+0.616i)T+11iT2 1 + (0.616 + 0.616i)T + 11iT^{2}
17 1+(1.80−3.11i)T+(−8.5−14.7i)T2 1 + (1.80 - 3.11i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.90−1.31i)T+(16.4−9.5i)T2 1 + (4.90 - 1.31i)T + (16.4 - 9.5i)T^{2}
23 1+(−1.86+3.23i)T+(−11.5−19.9i)T2 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2}
29 1−8.59iT−29T2 1 - 8.59iT - 29T^{2}
31 1+(−1.33+4.96i)T+(−26.8−15.5i)T2 1 + (-1.33 + 4.96i)T + (-26.8 - 15.5i)T^{2}
37 1+(5.77+1.54i)T+(32.0+18.5i)T2 1 + (5.77 + 1.54i)T + (32.0 + 18.5i)T^{2}
41 1+(−2.82−10.5i)T+(−35.5+20.5i)T2 1 + (-2.82 - 10.5i)T + (-35.5 + 20.5i)T^{2}
43 1+(−5.48+3.16i)T+(21.5−37.2i)T2 1 + (-5.48 + 3.16i)T + (21.5 - 37.2i)T^{2}
47 1+(−6.10+1.63i)T+(40.7−23.5i)T2 1 + (-6.10 + 1.63i)T + (40.7 - 23.5i)T^{2}
53 1−6.73iT−53T2 1 - 6.73iT - 53T^{2}
59 1+(3.89+3.89i)T+59iT2 1 + (3.89 + 3.89i)T + 59iT^{2}
61 1+(3.18+5.51i)T+(−30.5+52.8i)T2 1 + (3.18 + 5.51i)T + (-30.5 + 52.8i)T^{2}
67 1+(7.99−2.14i)T+(58.0−33.5i)T2 1 + (7.99 - 2.14i)T + (58.0 - 33.5i)T^{2}
71 1+(1.65+6.19i)T+(−61.4+35.5i)T2 1 + (1.65 + 6.19i)T + (-61.4 + 35.5i)T^{2}
73 1+(−10.6+10.6i)T−73iT2 1 + (-10.6 + 10.6i)T - 73iT^{2}
79 1+(1.30−2.26i)T+(−39.5−68.4i)T2 1 + (1.30 - 2.26i)T + (-39.5 - 68.4i)T^{2}
83 1+(−2.96−11.0i)T+(−71.8+41.5i)T2 1 + (-2.96 - 11.0i)T + (-71.8 + 41.5i)T^{2}
89 1+(−2.10+7.84i)T+(−77.0−44.5i)T2 1 + (-2.10 + 7.84i)T + (-77.0 - 44.5i)T^{2}
97 1+(4.37−16.3i)T+(−84.0−48.5i)T2 1 + (4.37 - 16.3i)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

−12.22792732146792045695025667891, −10.86352981788142585525049086243, −10.57563790095798019752726276539, −9.075173008670494226460023339957, −7.956942369715327797262414112027, −6.60078136649432439358627436650, −5.76426011015337419441678555328, −4.84884714509496938914410382114, −2.58719998429962459162531902816, −1.73660209323026482743477279687, 2.32705024925811297536814991884, 4.37252847207909906343890767438, 5.02568661355280824123589429612, 5.93776891768198565088241423186, 7.19386216674122025712352325849, 8.661858524750460977471696250755, 9.450182168605179404831457886954, 10.42807411469472661708284037998, 11.39813453206621867688886799395, 12.40564105162204584446888190964

Graph of the ZZ-function along the critical line