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Properties

Label 2-234-117.110-c1-0-9
Degree 22
Conductor 234234
Sign 0.620+0.784i0.620 + 0.784i
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  + (−0.258 + 0.965i)2-s + (0.675 − 1.59i)3-s + (−0.866 − 0.499i)4-s + (0.435 − 1.62i)5-s + (1.36 + 1.06i)6-s + (−0.212 − 0.212i)7-s + (0.707 − 0.707i)8-s + (−2.08 − 2.15i)9-s + (1.45 + 0.842i)10-s + (−1.53 − 0.411i)11-s + (−1.38 + 1.04i)12-s + (2.24 − 2.82i)13-s + (0.260 − 0.150i)14-s + (−2.29 − 1.79i)15-s + (0.500 + 0.866i)16-s + (−1.26 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.390 − 0.920i)3-s + (−0.433 − 0.249i)4-s + (0.194 − 0.727i)5-s + (0.557 + 0.434i)6-s + (−0.0803 − 0.0803i)7-s + (0.249 − 0.249i)8-s + (−0.695 − 0.718i)9-s + (0.461 + 0.266i)10-s + (−0.462 − 0.124i)11-s + (−0.399 + 0.301i)12-s + (0.622 − 0.782i)13-s + (0.0695 − 0.0401i)14-s + (−0.593 − 0.463i)15-s + (0.125 + 0.216i)16-s + (−0.306 − 0.531i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.620+0.784i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.620+0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.620+0.784i0.620 + 0.784i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(227,⋅)\chi_{234} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.620+0.784i)(2,\ 234,\ (\ :1/2),\ 0.620 + 0.784i)

Particular Values

L(1)L(1) ≈\approx 1.06800−0.517165i1.06800 - 0.517165i
L(12)L(\frac12) ≈\approx 1.06800−0.517165i1.06800 - 0.517165i
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+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
3 1+(−0.675+1.59i)T 1 + (-0.675 + 1.59i)T
13 1+(−2.24+2.82i)T 1 + (-2.24 + 2.82i)T
good5 1+(−0.435+1.62i)T+(−4.33−2.5i)T2 1 + (-0.435 + 1.62i)T + (-4.33 - 2.5i)T^{2}
7 1+(0.212+0.212i)T+7iT2 1 + (0.212 + 0.212i)T + 7iT^{2}
11 1+(1.53+0.411i)T+(9.52+5.5i)T2 1 + (1.53 + 0.411i)T + (9.52 + 5.5i)T^{2}
17 1+(1.26+2.19i)T+(−8.5+14.7i)T2 1 + (1.26 + 2.19i)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 1−1.32T+23T2 1 - 1.32T + 23T^{2}
29 1+(6.48−3.74i)T+(14.5−25.1i)T2 1 + (6.48 - 3.74i)T + (14.5 - 25.1i)T^{2}
31 1+(−10.3−2.78i)T+(26.8+15.5i)T2 1 + (-10.3 - 2.78i)T + (26.8 + 15.5i)T^{2}
37 1+(2.37−0.637i)T+(32.0−18.5i)T2 1 + (2.37 - 0.637i)T + (32.0 - 18.5i)T^{2}
41 1+(−2.66−2.66i)T+41iT2 1 + (-2.66 - 2.66i)T + 41iT^{2}
43 1−7.42iT−43T2 1 - 7.42iT - 43T^{2}
47 1+(−0.928−3.46i)T+(−40.7+23.5i)T2 1 + (-0.928 - 3.46i)T + (-40.7 + 23.5i)T^{2}
53 1+3.63iT−53T2 1 + 3.63iT - 53T^{2}
59 1+(−0.187−0.699i)T+(−51.0+29.5i)T2 1 + (-0.187 - 0.699i)T + (-51.0 + 29.5i)T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1+(0.690−0.690i)T−67iT2 1 + (0.690 - 0.690i)T - 67iT^{2}
71 1+(1.84−6.87i)T+(−61.4−35.5i)T2 1 + (1.84 - 6.87i)T + (-61.4 - 35.5i)T^{2}
73 1+(−3.38−3.38i)T+73iT2 1 + (-3.38 - 3.38i)T + 73iT^{2}
79 1+(3.69−6.40i)T+(−39.5−68.4i)T2 1 + (3.69 - 6.40i)T + (-39.5 - 68.4i)T^{2}
83 1+(−11.8+3.18i)T+(71.8−41.5i)T2 1 + (-11.8 + 3.18i)T + (71.8 - 41.5i)T^{2}
89 1+(0.512+1.91i)T+(−77.0+44.5i)T2 1 + (0.512 + 1.91i)T + (-77.0 + 44.5i)T^{2}
97 1+(−4.41+4.41i)T−97iT2 1 + (-4.41 + 4.41i)T - 97iT^{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

−12.35575588878714697886855304262, −11.14781638867457846198232118326, −9.772534097351300053382872159902, −8.812699695459869726387991215125, −8.065752006577822007651953550406, −7.15020490882589427728412268428, −6.00111073844031544179447247219, −5.01561701739341805436251299586, −3.13447659633336656521644037699, −1.10128160496247438849510158010, 2.35646302185415736984197050492, 3.49576643217525880573789022461, 4.65158292312950449913506690440, 6.06668830383665271245798989083, 7.56808726116132783377315438885, 8.746180394114616896432797094761, 9.543232632108439803614986697371, 10.46442096218553048608134779417, 11.07179936694065348935784046540, 12.03397324109907918189085743663

Graph of the ZZ-function along the critical line