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Properties

Label 2-234-117.25-c1-0-1
Degree 22
Conductor 234234
Sign 0.728−0.684i0.728 - 0.684i
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  + (−0.866 − 0.5i)2-s + (−0.579 − 1.63i)3-s + (0.499 + 0.866i)4-s + (−3.32 + 1.91i)5-s + (−0.314 + 1.70i)6-s + (2.91 + 1.68i)7-s − 0.999i·8-s + (−2.32 + 1.89i)9-s + 3.83·10-s + (1.72 + 0.995i)11-s + (1.12 − 1.31i)12-s + (−0.985 + 3.46i)13-s + (−1.68 − 2.91i)14-s + (5.05 + 4.31i)15-s + (−0.5 + 0.866i)16-s + 2.60·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.334 − 0.942i)3-s + (0.249 + 0.433i)4-s + (−1.48 + 0.857i)5-s + (−0.128 + 0.695i)6-s + (1.10 + 0.636i)7-s − 0.353i·8-s + (−0.776 + 0.630i)9-s + 1.21·10-s + (0.520 + 0.300i)11-s + (0.324 − 0.380i)12-s + (−0.273 + 0.961i)13-s + (−0.450 − 0.779i)14-s + (1.30 + 1.11i)15-s + (−0.125 + 0.216i)16-s + 0.630·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.728−0.684i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.728−0.684i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.728−0.684i0.728 - 0.684i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(25,⋅)\chi_{234} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.728−0.684i)(2,\ 234,\ (\ :1/2),\ 0.728 - 0.684i)

Particular Values

L(1)L(1) ≈\approx 0.556911+0.220660i0.556911 + 0.220660i
L(12)L(\frac12) ≈\approx 0.556911+0.220660i0.556911 + 0.220660i
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(0.579+1.63i)T 1 + (0.579 + 1.63i)T
13 1+(0.985−3.46i)T 1 + (0.985 - 3.46i)T
good5 1+(3.32−1.91i)T+(2.5−4.33i)T2 1 + (3.32 - 1.91i)T + (2.5 - 4.33i)T^{2}
7 1+(−2.91−1.68i)T+(3.5+6.06i)T2 1 + (-2.91 - 1.68i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.72−0.995i)T+(5.5+9.52i)T2 1 + (-1.72 - 0.995i)T + (5.5 + 9.52i)T^{2}
17 1−2.60T+17T2 1 - 2.60T + 17T^{2}
19 1+1.99iT−19T2 1 + 1.99iT - 19T^{2}
23 1+(−2.13−3.70i)T+(−11.5+19.9i)T2 1 + (-2.13 - 3.70i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.37−7.57i)T+(−14.5−25.1i)T2 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2}
31 1+(4.57−2.64i)T+(15.5−26.8i)T2 1 + (4.57 - 2.64i)T + (15.5 - 26.8i)T^{2}
37 1−5.08iT−37T2 1 - 5.08iT - 37T^{2}
41 1+(−7.13+4.12i)T+(20.5−35.5i)T2 1 + (-7.13 + 4.12i)T + (20.5 - 35.5i)T^{2}
43 1+(5.13−8.88i)T+(−21.5−37.2i)T2 1 + (5.13 - 8.88i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.22+2.44i)T+(23.5+40.7i)T2 1 + (4.22 + 2.44i)T + (23.5 + 40.7i)T^{2}
53 1+9.16T+53T2 1 + 9.16T + 53T^{2}
59 1+(−6.33+3.65i)T+(29.5−51.0i)T2 1 + (-6.33 + 3.65i)T + (29.5 - 51.0i)T^{2}
61 1+(−5.63+9.76i)T+(−30.5−52.8i)T2 1 + (-5.63 + 9.76i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.90−2.83i)T+(33.5−58.0i)T2 1 + (4.90 - 2.83i)T + (33.5 - 58.0i)T^{2}
71 1−1.94iT−71T2 1 - 1.94iT - 71T^{2}
73 1+8.41iT−73T2 1 + 8.41iT - 73T^{2}
79 1+(−2.97+5.15i)T+(−39.5−68.4i)T2 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2}
83 1+(−2.03−1.17i)T+(41.5+71.8i)T2 1 + (-2.03 - 1.17i)T + (41.5 + 71.8i)T^{2}
89 1−6.28iT−89T2 1 - 6.28iT - 89T^{2}
97 1+(−8.92−5.15i)T+(48.5+84.0i)T2 1 + (-8.92 - 5.15i)T + (48.5 + 84.0i)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

−11.85375008041248590083250151878, −11.46805850620971836630567555421, −10.88546067082397737785966240846, −9.179085059470702808828981315201, −8.145465452074967399843340879825, −7.42519876872847634014647395313, −6.70358136859963011518104733546, −4.93796269094145841954998369654, −3.31602315052713877078848801825, −1.73255538269655530829488288100, 0.66017336061119900310996449685, 3.74742622915692270406001412105, 4.63925065257298138073925738083, 5.68583462026333758853900065066, 7.50626569140115857001132887499, 8.115674711576626579463557744032, 8.975295680086208942474541887875, 10.16479966413188274683725354881, 11.16516061912183657198291421642, 11.63255736193953416874582531761

Graph of the ZZ-function along the critical line