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Properties

Label 2-832-13.10-c1-0-22
Degree 22
Conductor 832832
Sign −0.252+0.967i-0.252 + 0.967i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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 + 1.5i)3-s − 3.46i·5-s + (−2.59 − 1.5i)7-s + (−4.33 + 2.5i)11-s + (1 + 3.46i)13-s + (5.19 − 2.99i)15-s + (3.5 − 6.06i)17-s + (−4.33 − 2.5i)19-s − 5.19i·21-s + (−2.59 − 4.5i)23-s − 6.99·25-s + 5.19·27-s + (−2.5 − 4.33i)29-s − 2i·31-s + (−7.5 − 4.33i)33-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s − 1.54i·5-s + (−0.981 − 0.566i)7-s + (−1.30 + 0.753i)11-s + (0.277 + 0.960i)13-s + (1.34 − 0.774i)15-s + (0.848 − 1.47i)17-s + (−0.993 − 0.573i)19-s − 1.13i·21-s + (−0.541 − 0.938i)23-s − 1.39·25-s + 1.00·27-s + (−0.464 − 0.804i)29-s − 0.359i·31-s + (−1.30 − 0.753i)33-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.252+0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.252+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.252+0.967i-0.252 + 0.967i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(257,⋅)\chi_{832} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), −0.252+0.967i)(2,\ 832,\ (\ :1/2),\ -0.252 + 0.967i)

Particular Values

L(1)L(1) ≈\approx 0.644801−0.834766i0.644801 - 0.834766i
L(12)L(\frac12) ≈\approx 0.644801−0.834766i0.644801 - 0.834766i
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
13 1+(−1−3.46i)T 1 + (-1 - 3.46i)T
good3 1+(−0.866−1.5i)T+(−1.5+2.59i)T2 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2}
5 1+3.46iT−5T2 1 + 3.46iT - 5T^{2}
7 1+(2.59+1.5i)T+(3.5+6.06i)T2 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2}
11 1+(4.33−2.5i)T+(5.5−9.52i)T2 1 + (4.33 - 2.5i)T + (5.5 - 9.52i)T^{2}
17 1+(−3.5+6.06i)T+(−8.5−14.7i)T2 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.33+2.5i)T+(9.5+16.4i)T2 1 + (4.33 + 2.5i)T + (9.5 + 16.4i)T^{2}
23 1+(2.59+4.5i)T+(−11.5+19.9i)T2 1 + (2.59 + 4.5i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.5+4.33i)T+(−14.5+25.1i)T2 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+(4.5−2.59i)T+(18.5−32.0i)T2 1 + (4.5 - 2.59i)T + (18.5 - 32.0i)T^{2}
41 1+(1.5−0.866i)T+(20.5−35.5i)T2 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.59+4.5i)T+(−21.5−37.2i)T2 1 + (-2.59 + 4.5i)T + (-21.5 - 37.2i)T^{2}
47 1+4iT−47T2 1 + 4iT - 47T^{2}
53 1+4T+53T2 1 + 4T + 53T^{2}
59 1+(−6.06−3.5i)T+(29.5+51.0i)T2 1 + (-6.06 - 3.5i)T + (29.5 + 51.0i)T^{2}
61 1+(−1.5+2.59i)T+(−30.5−52.8i)T2 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.59−1.5i)T+(33.5−58.0i)T2 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2}
71 1+(−6.06−3.5i)T+(35.5+61.4i)T2 1 + (-6.06 - 3.5i)T + (35.5 + 61.4i)T^{2}
73 1−3.46iT−73T2 1 - 3.46iT - 73T^{2}
79 1−3.46T+79T2 1 - 3.46T + 79T^{2}
83 1+14iT−83T2 1 + 14iT - 83T^{2}
89 1+(−1.5+0.866i)T+(44.5−77.0i)T2 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2}
97 1+(−7.5−4.33i)T+(48.5+84.0i)T2 1 + (-7.5 - 4.33i)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

−9.877998204908862386110032871315, −9.207234192835200423614294617761, −8.565284769052063164123669039898, −7.52750766774822474121386055996, −6.51816711380605981272777082346, −5.14918830804095099354067444517, −4.53977221518602306500339049214, −3.73383518618122464812277916686, −2.37212028449560547990958365689, −0.45712327582775890214975589543, 1.97046832394155394436606828951, 3.03507686231489644484628742773, 3.46701889614255901547158827532, 5.64951691037786993164926234045, 6.12544295282820725141904832160, 7.08719998194854200612340886851, 7.943818392258182514897943923733, 8.376585893064068134046467771425, 9.825858627751316530929121688513, 10.60195774873160405284938116000

Graph of the ZZ-function along the critical line