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Properties

Label 2-234-117.32-c1-0-10
Degree 22
Conductor 234234
Sign 0.938+0.345i0.938 + 0.345i
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 + (1.67 + 0.444i)3-s + 1.00i·4-s + (3.83 − 1.02i)5-s + (−0.869 − 1.49i)6-s + (−1.55 + 0.415i)7-s + (0.707 − 0.707i)8-s + (2.60 + 1.48i)9-s + (−3.43 − 1.98i)10-s + (−3.50 + 3.50i)11-s + (−0.444 + 1.67i)12-s + (−1.03 − 3.45i)13-s + (1.39 + 0.803i)14-s + (6.87 − 0.0145i)15-s − 1.00·16-s + (0.584 + 1.01i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.966 + 0.256i)3-s + 0.500i·4-s + (1.71 − 0.459i)5-s + (−0.354 − 0.611i)6-s + (−0.586 + 0.157i)7-s + (0.250 − 0.250i)8-s + (0.868 + 0.496i)9-s + (−1.08 − 0.627i)10-s + (−1.05 + 1.05i)11-s + (−0.128 + 0.483i)12-s + (−0.288 − 0.957i)13-s + (0.371 + 0.214i)14-s + (1.77 − 0.00375i)15-s − 0.250·16-s + (0.141 + 0.245i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.938+0.345i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.938+0.345i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.938+0.345i0.938 + 0.345i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(149,⋅)\chi_{234} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.938+0.345i)(2,\ 234,\ (\ :1/2),\ 0.938 + 0.345i)

Particular Values

L(1)L(1) ≈\approx 1.48255−0.264355i1.48255 - 0.264355i
L(12)L(\frac12) ≈\approx 1.48255−0.264355i1.48255 - 0.264355i
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+(−1.67−0.444i)T 1 + (-1.67 - 0.444i)T
13 1+(1.03+3.45i)T 1 + (1.03 + 3.45i)T
good5 1+(−3.83+1.02i)T+(4.33−2.5i)T2 1 + (-3.83 + 1.02i)T + (4.33 - 2.5i)T^{2}
7 1+(1.55−0.415i)T+(6.06−3.5i)T2 1 + (1.55 - 0.415i)T + (6.06 - 3.5i)T^{2}
11 1+(3.50−3.50i)T−11iT2 1 + (3.50 - 3.50i)T - 11iT^{2}
17 1+(−0.584−1.01i)T+(−8.5+14.7i)T2 1 + (-0.584 - 1.01i)T + (-8.5 + 14.7i)T^{2}
19 1+(4.16+1.11i)T+(16.4+9.5i)T2 1 + (4.16 + 1.11i)T + (16.4 + 9.5i)T^{2}
23 1+(−1.63−2.83i)T+(−11.5+19.9i)T2 1 + (-1.63 - 2.83i)T + (-11.5 + 19.9i)T^{2}
29 1+7.50iT−29T2 1 + 7.50iT - 29T^{2}
31 1+(1.94+7.25i)T+(−26.8+15.5i)T2 1 + (1.94 + 7.25i)T + (-26.8 + 15.5i)T^{2}
37 1+(4.28−1.14i)T+(32.0−18.5i)T2 1 + (4.28 - 1.14i)T + (32.0 - 18.5i)T^{2}
41 1+(1.44−5.41i)T+(−35.5−20.5i)T2 1 + (1.44 - 5.41i)T + (-35.5 - 20.5i)T^{2}
43 1+(0.770+0.444i)T+(21.5+37.2i)T2 1 + (0.770 + 0.444i)T + (21.5 + 37.2i)T^{2}
47 1+(−2.43−0.653i)T+(40.7+23.5i)T2 1 + (-2.43 - 0.653i)T + (40.7 + 23.5i)T^{2}
53 1−6.60iT−53T2 1 - 6.60iT - 53T^{2}
59 1+(−5.81+5.81i)T−59iT2 1 + (-5.81 + 5.81i)T - 59iT^{2}
61 1+(7.05−12.2i)T+(−30.5−52.8i)T2 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.19+0.855i)T+(58.0+33.5i)T2 1 + (3.19 + 0.855i)T + (58.0 + 33.5i)T^{2}
71 1+(−0.942+3.51i)T+(−61.4−35.5i)T2 1 + (-0.942 + 3.51i)T + (-61.4 - 35.5i)T^{2}
73 1+(3.33+3.33i)T+73iT2 1 + (3.33 + 3.33i)T + 73iT^{2}
79 1+(1.16+2.02i)T+(−39.5+68.4i)T2 1 + (1.16 + 2.02i)T + (-39.5 + 68.4i)T^{2}
83 1+(2.82−10.5i)T+(−71.8−41.5i)T2 1 + (2.82 - 10.5i)T + (-71.8 - 41.5i)T^{2}
89 1+(1.78+6.65i)T+(−77.0+44.5i)T2 1 + (1.78 + 6.65i)T + (-77.0 + 44.5i)T^{2}
97 1+(1.65+6.17i)T+(−84.0+48.5i)T2 1 + (1.65 + 6.17i)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

−12.52239603647713144419858402923, −10.61484545842654447147265775755, −9.893900900267100323883326787525, −9.551592097861042272063430286599, −8.472494198252959903913748272585, −7.42597538419669795384917964109, −5.90974694288548258249387328447, −4.64737747586945621350229126505, −2.81439928967093052849388242729, −1.98930521193740247269392832466, 1.92020018230108385184546902031, 3.11197061608121771681908992683, 5.24648075824000802976927669520, 6.45940698429993905504441488995, 7.05621346328667778594614068241, 8.568454091458252745490078295210, 9.140805977313498920091908534310, 10.17267753568737865203088541055, 10.68117167944282690393444898131, 12.66829401593755248949645317042

Graph of the ZZ-function along the critical line