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Properties

Label 2-234-117.11-c1-0-1
Degree 22
Conductor 234234
Sign −0.122−0.992i-0.122 - 0.992i
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 + (−1.04 + 1.38i)3-s − 1.00i·4-s + (1.51 + 0.406i)5-s + (−0.243 − 1.71i)6-s + (4.52 + 1.21i)7-s + (0.707 + 0.707i)8-s + (−0.834 − 2.88i)9-s + (−1.36 + 0.785i)10-s + (−1.08 − 1.08i)11-s + (1.38 + 1.04i)12-s + (−1.01 + 3.46i)13-s + (−4.05 + 2.34i)14-s + (−2.14 + 1.67i)15-s − 1.00·16-s + (−2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.600 + 0.799i)3-s − 0.500i·4-s + (0.678 + 0.181i)5-s + (−0.0993 − 0.700i)6-s + (1.71 + 0.458i)7-s + (0.250 + 0.250i)8-s + (−0.278 − 0.960i)9-s + (−0.430 + 0.248i)10-s + (−0.326 − 0.326i)11-s + (0.399 + 0.300i)12-s + (−0.280 + 0.959i)13-s + (−1.08 + 0.626i)14-s + (−0.552 + 0.433i)15-s − 0.250·16-s + (−0.725 + 1.25i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.122−0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.122−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.122−0.992i-0.122 - 0.992i
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), −0.122−0.992i)(2,\ 234,\ (\ :1/2),\ -0.122 - 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.651392+0.736894i0.651392 + 0.736894i
L(12)L(\frac12) ≈\approx 0.651392+0.736894i0.651392 + 0.736894i
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+(1.04−1.38i)T 1 + (1.04 - 1.38i)T
13 1+(1.01−3.46i)T 1 + (1.01 - 3.46i)T
good5 1+(−1.51−0.406i)T+(4.33+2.5i)T2 1 + (-1.51 - 0.406i)T + (4.33 + 2.5i)T^{2}
7 1+(−4.52−1.21i)T+(6.06+3.5i)T2 1 + (-4.52 - 1.21i)T + (6.06 + 3.5i)T^{2}
11 1+(1.08+1.08i)T+11iT2 1 + (1.08 + 1.08i)T + 11iT^{2}
17 1+(2.99−5.18i)T+(−8.5−14.7i)T2 1 + (2.99 - 5.18i)T + (-8.5 - 14.7i)T^{2}
19 1+(−5.70+1.52i)T+(16.4−9.5i)T2 1 + (-5.70 + 1.52i)T + (16.4 - 9.5i)T^{2}
23 1+(2.48−4.30i)T+(−11.5−19.9i)T2 1 + (2.48 - 4.30i)T + (-11.5 - 19.9i)T^{2}
29 1+6.86iT−29T2 1 + 6.86iT - 29T^{2}
31 1+(0.271−1.01i)T+(−26.8−15.5i)T2 1 + (0.271 - 1.01i)T + (-26.8 - 15.5i)T^{2}
37 1+(6.94+1.86i)T+(32.0+18.5i)T2 1 + (6.94 + 1.86i)T + (32.0 + 18.5i)T^{2}
41 1+(−1.34−5.00i)T+(−35.5+20.5i)T2 1 + (-1.34 - 5.00i)T + (-35.5 + 20.5i)T^{2}
43 1+(−10.6+6.12i)T+(21.5−37.2i)T2 1 + (-10.6 + 6.12i)T + (21.5 - 37.2i)T^{2}
47 1+(−6.14+1.64i)T+(40.7−23.5i)T2 1 + (-6.14 + 1.64i)T + (40.7 - 23.5i)T^{2}
53 1+2.78iT−53T2 1 + 2.78iT - 53T^{2}
59 1+(4.22+4.22i)T+59iT2 1 + (4.22 + 4.22i)T + 59iT^{2}
61 1+(2.18+3.78i)T+(−30.5+52.8i)T2 1 + (2.18 + 3.78i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.41−1.45i)T+(58.0−33.5i)T2 1 + (5.41 - 1.45i)T + (58.0 - 33.5i)T^{2}
71 1+(−0.0745−0.278i)T+(−61.4+35.5i)T2 1 + (-0.0745 - 0.278i)T + (-61.4 + 35.5i)T^{2}
73 1+(−0.964+0.964i)T−73iT2 1 + (-0.964 + 0.964i)T - 73iT^{2}
79 1+(0.0673−0.116i)T+(−39.5−68.4i)T2 1 + (0.0673 - 0.116i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.00+11.2i)T+(−71.8+41.5i)T2 1 + (3.00 + 11.2i)T + (-71.8 + 41.5i)T^{2}
89 1+(−1.46+5.45i)T+(−77.0−44.5i)T2 1 + (-1.46 + 5.45i)T + (-77.0 - 44.5i)T^{2}
97 1+(1.72−6.44i)T+(−84.0−48.5i)T2 1 + (1.72 - 6.44i)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

−11.97898780026842100518530102154, −11.30866747513275190443572961755, −10.49889250289969143507377803743, −9.482942756561868129415388615955, −8.665519260380290080244542803938, −7.54921427566294499930724601645, −6.08440909848499261128061119431, −5.37815506349575705795303924486, −4.29237360266272092147884442672, −1.91026515153173549653959407435, 1.16800219606034841704185782218, 2.41708590897414193419382295711, 4.76003520928271690539308840980, 5.55910925573866853985781879658, 7.29627203475762711189539078400, 7.76897684586577307551295440983, 8.959853539289331088610048337530, 10.29745239419458163420899745180, 10.95541718596267555237442021920, 11.84098939060121549316915878155

Graph of the ZZ-function along the critical line