↑

Properties

Label 2-1512-21.17-c1-0-24
Degree 22
Conductor 15121512
Sign −0.261+0.965i-0.261 + 0.965i
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 − 1.95i)5-s + (2.62 − 0.349i)7-s + (−2.61 − 1.50i)11-s + 3.29i·13-s + (3.31 − 5.73i)17-s + (−0.605 + 0.349i)19-s + (1.73 − i)23-s + (−0.0520 + 0.0902i)25-s + 1.45i·29-s + (0.323 + 0.186i)31-s + (−3.64 − 4.73i)35-s + (−5.63 − 9.75i)37-s − 2.10·41-s − 43-s + (−3.06 − 5.30i)47-s + ⋯
L(s)  = 1  + (−0.505 − 0.875i)5-s + (0.991 − 0.132i)7-s + (−0.787 − 0.454i)11-s + 0.913i·13-s + (0.803 − 1.39i)17-s + (−0.138 + 0.0801i)19-s + (0.361 − 0.208i)23-s + (−0.0104 + 0.0180i)25-s + 0.270i·29-s + (0.0580 + 0.0335i)31-s + (−0.616 − 0.800i)35-s + (−0.925 − 1.60i)37-s − 0.328·41-s − 0.152·43-s + (−0.446 − 0.773i)47-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((−0.261+0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s+1/2) L(s)=((−0.261+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: −0.261+0.965i-0.261 + 0.965i
Analytic conductor: 12.073312.0733
Root analytic conductor: 3.474673.47467
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1512(1025,⋅)\chi_{1512} (1025, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :1/2), −0.261+0.965i)(2,\ 1512,\ (\ :1/2),\ -0.261 + 0.965i)

Particular Values

L(1)L(1) ≈\approx 1.3503964801.350396480
L(12)L(\frac12) ≈\approx 1.3503964801.350396480
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 1
3 1 1
7 1+(−2.62+0.349i)T 1 + (-2.62 + 0.349i)T
good5 1+(1.12+1.95i)T+(−2.5+4.33i)T2 1 + (1.12 + 1.95i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.61+1.50i)T+(5.5+9.52i)T2 1 + (2.61 + 1.50i)T + (5.5 + 9.52i)T^{2}
13 1−3.29iT−13T2 1 - 3.29iT - 13T^{2}
17 1+(−3.31+5.73i)T+(−8.5−14.7i)T2 1 + (-3.31 + 5.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.605−0.349i)T+(9.5−16.4i)T2 1 + (0.605 - 0.349i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.73+i)T+(11.5−19.9i)T2 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2}
29 1−1.45iT−29T2 1 - 1.45iT - 29T^{2}
31 1+(−0.323−0.186i)T+(15.5+26.8i)T2 1 + (-0.323 - 0.186i)T + (15.5 + 26.8i)T^{2}
37 1+(5.63+9.75i)T+(−18.5+32.0i)T2 1 + (5.63 + 9.75i)T + (-18.5 + 32.0i)T^{2}
41 1+2.10T+41T2 1 + 2.10T + 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1+(3.06+5.30i)T+(−23.5+40.7i)T2 1 + (3.06 + 5.30i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.0299+0.0173i)T+(26.5+45.8i)T2 1 + (0.0299 + 0.0173i)T + (26.5 + 45.8i)T^{2}
59 1+(2.61−4.52i)T+(−29.5−51.0i)T2 1 + (2.61 - 4.52i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.36−2.52i)T+(30.5−52.8i)T2 1 + (4.36 - 2.52i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.34+2.32i)T+(−33.5−58.0i)T2 1 + (-1.34 + 2.32i)T + (-33.5 - 58.0i)T^{2}
71 1+10.4iT−71T2 1 + 10.4iT - 71T^{2}
73 1+(11.7+6.80i)T+(36.5+63.2i)T2 1 + (11.7 + 6.80i)T + (36.5 + 63.2i)T^{2}
79 1+(−7.52−13.0i)T+(−39.5+68.4i)T2 1 + (-7.52 - 13.0i)T + (-39.5 + 68.4i)T^{2}
83 1+6.58T+83T2 1 + 6.58T + 83T^{2}
89 1+(7.50+12.9i)T+(−44.5+77.0i)T2 1 + (7.50 + 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1+6.09iT−97T2 1 + 6.09iT - 97T^{2}
show more
show less
   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

−8.959449413188930618248781823119, −8.559873189220197939703644039925, −7.64359869397836609102769920587, −7.09870257394006704013261215559, −5.70733495156937855776799169167, −4.96323454128858322947070149510, −4.38852000963870285184725583903, −3.17777457596743787760662344427, −1.85137024260003182634226210711, −0.55142050024298411466691104180, 1.50100162175444323084200927107, 2.77728962416655073747544244017, 3.61356882510127213520200859461, 4.77044444982280116020478692238, 5.52157940096113223088287085705, 6.51737260826467438102783418485, 7.54349808816161589226465207238, 7.957919253563786325469736642871, 8.657572058512246247060385310108, 10.01274939022891581126960329242

Graph of the ZZ-function along the critical line