↑

Properties

Label 2-1470-35.27-c1-0-27
Degree 22
Conductor 14701470
Sign −0.112−0.993i-0.112 - 0.993i
Analytic cond. 11.738011.7380
Root an. cond. 3.426073.42607
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  + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (2.14 + 0.625i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (1.07 + 1.96i)10-s + 2.07·11-s + (−0.707 + 0.707i)12-s + (−0.326 − 0.326i)13-s + (1.07 + 1.96i)15-s − 1.00·16-s + (−1.26 + 1.26i)17-s + (−0.707 + 0.707i)18-s + 4.37·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.960 + 0.279i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.340 + 0.619i)10-s + 0.625·11-s + (−0.204 + 0.204i)12-s + (−0.0906 − 0.0906i)13-s + (0.277 + 0.506i)15-s − 0.250·16-s + (−0.307 + 0.307i)17-s + (−0.166 + 0.166i)18-s + 1.00·19-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=((−0.112−0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+1/2) L(s)=((−0.112−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −0.112−0.993i-0.112 - 0.993i
Analytic conductor: 11.738011.7380
Root analytic conductor: 3.426073.42607
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1470(97,⋅)\chi_{1470} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1470, ( :1/2), −0.112−0.993i)(2,\ 1470,\ (\ :1/2),\ -0.112 - 0.993i)

Particular Values

L(1)L(1) ≈\approx 3.0893673553.089367355
L(12)L(\frac12) ≈\approx 3.0893673553.089367355
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1+(−2.14−0.625i)T 1 + (-2.14 - 0.625i)T
7 1 1
good11 1−2.07T+11T2 1 - 2.07T + 11T^{2}
13 1+(0.326+0.326i)T+13iT2 1 + (0.326 + 0.326i)T + 13iT^{2}
17 1+(1.26−1.26i)T−17iT2 1 + (1.26 - 1.26i)T - 17iT^{2}
19 1−4.37T+19T2 1 - 4.37T + 19T^{2}
23 1+(−0.635+0.635i)T−23iT2 1 + (-0.635 + 0.635i)T - 23iT^{2}
29 1+0.0288iT−29T2 1 + 0.0288iT - 29T^{2}
31 1−8.03iT−31T2 1 - 8.03iT - 31T^{2}
37 1+(8.07+8.07i)T+37iT2 1 + (8.07 + 8.07i)T + 37iT^{2}
41 1−10.6iT−41T2 1 - 10.6iT - 41T^{2}
43 1+(−2.50+2.50i)T−43iT2 1 + (-2.50 + 2.50i)T - 43iT^{2}
47 1+(0.525−0.525i)T−47iT2 1 + (0.525 - 0.525i)T - 47iT^{2}
53 1+(−7.22+7.22i)T−53iT2 1 + (-7.22 + 7.22i)T - 53iT^{2}
59 1+13.4T+59T2 1 + 13.4T + 59T^{2}
61 1+9.84iT−61T2 1 + 9.84iT - 61T^{2}
67 1+(3.33+3.33i)T+67iT2 1 + (3.33 + 3.33i)T + 67iT^{2}
71 1−2.27T+71T2 1 - 2.27T + 71T^{2}
73 1+(−8.14−8.14i)T+73iT2 1 + (-8.14 - 8.14i)T + 73iT^{2}
79 1+8.01iT−79T2 1 + 8.01iT - 79T^{2}
83 1+(−4.26−4.26i)T+83iT2 1 + (-4.26 - 4.26i)T + 83iT^{2}
89 1+0.0197T+89T2 1 + 0.0197T + 89T^{2}
97 1+(5.65−5.65i)T−97iT2 1 + (5.65 - 5.65i)T - 97iT^{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

−9.527124602525643416067904109427, −9.019356539462861261111259110896, −8.127373360425713957528790355315, −7.09776817871288330377112189110, −6.48346043833477438474682014351, −5.51418490620302278032146288054, −4.85628310214992640960753412337, −3.70259560213253441445298947157, −2.90247797663162993361259649057, −1.67906361394002912750422403207, 1.08986162554754029698472182176, 2.08325920384585867080373852967, 3.01917837625439411689405690333, 4.10733616053532646661925610449, 5.13156844877534222792469494364, 5.91995362619103060403560670795, 6.73928073527114198384043037334, 7.58956717137292342094399296712, 8.803299635644904553964835911768, 9.294921064168835868287563984562

Graph of the ZZ-function along the critical line