↑

Properties

Label 2-832-52.31-c1-0-21
Degree 22
Conductor 832832
Sign −0.916+0.399i-0.916 + 0.399i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−1.58 + 1.58i)5-s + (−1.58 + 1.58i)7-s + 2·9-s + (−2.16 + 2.16i)11-s + (−0.418 − 3.58i)13-s + (1.58 + 1.58i)15-s − 5.32i·17-s + (−5.16 − 5.16i)19-s + (1.58 + 1.58i)21-s − 0.837·23-s − 5i·27-s − 5.16·29-s + (−5.16 − 5.16i)31-s + (2.16 + 2.16i)33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.707 + 0.707i)5-s + (−0.597 + 0.597i)7-s + 0.666·9-s + (−0.651 + 0.651i)11-s + (−0.116 − 0.993i)13-s + (0.408 + 0.408i)15-s − 1.29i·17-s + (−1.18 − 1.18i)19-s + (0.345 + 0.345i)21-s − 0.174·23-s − 0.962i·27-s − 0.958·29-s + (−0.927 − 0.927i)31-s + (0.376 + 0.376i)33-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.916+0.399i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.916+0.399i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.916+0.399i-0.916 + 0.399i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(447,⋅)\chi_{832} (447, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), −0.916+0.399i)(2,\ 832,\ (\ :1/2),\ -0.916 + 0.399i)

Particular Values

L(1)L(1) ≈\approx 0.0644345−0.309561i0.0644345 - 0.309561i
L(12)L(\frac12) ≈\approx 0.0644345−0.309561i0.0644345 - 0.309561i
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+(0.418+3.58i)T 1 + (0.418 + 3.58i)T
good3 1+iT−3T2 1 + iT - 3T^{2}
5 1+(1.58−1.58i)T−5iT2 1 + (1.58 - 1.58i)T - 5iT^{2}
7 1+(1.58−1.58i)T−7iT2 1 + (1.58 - 1.58i)T - 7iT^{2}
11 1+(2.16−2.16i)T−11iT2 1 + (2.16 - 2.16i)T - 11iT^{2}
17 1+5.32iT−17T2 1 + 5.32iT - 17T^{2}
19 1+(5.16+5.16i)T+19iT2 1 + (5.16 + 5.16i)T + 19iT^{2}
23 1+0.837T+23T2 1 + 0.837T + 23T^{2}
29 1+5.16T+29T2 1 + 5.16T + 29T^{2}
31 1+(5.16+5.16i)T+31iT2 1 + (5.16 + 5.16i)T + 31iT^{2}
37 1+(−0.418−0.418i)T+37iT2 1 + (-0.418 - 0.418i)T + 37iT^{2}
41 1+(1.16−1.16i)T−41iT2 1 + (1.16 - 1.16i)T - 41iT^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1+(2.74−2.74i)T−47iT2 1 + (2.74 - 2.74i)T - 47iT^{2}
53 1−9.48T+53T2 1 - 9.48T + 53T^{2}
59 1+(4−4i)T−59iT2 1 + (4 - 4i)T - 59iT^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+(−5.32−5.32i)T+67iT2 1 + (-5.32 - 5.32i)T + 67iT^{2}
71 1+(1.58+1.58i)T+71iT2 1 + (1.58 + 1.58i)T + 71iT^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1+15.4iT−79T2 1 + 15.4iT - 79T^{2}
83 1+(−12.1−12.1i)T+83iT2 1 + (-12.1 - 12.1i)T + 83iT^{2}
89 1+(9.16+9.16i)T+89iT2 1 + (9.16 + 9.16i)T + 89iT^{2}
97 1+(10.1−10.1i)T−97iT2 1 + (10.1 - 10.1i)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

−9.867372595472121538325236328496, −9.052569281431609077859926535143, −7.81172137108504189357303348753, −7.32074279751196202585049449167, −6.62174779970296582739392750507, −5.48019395920802573787781343012, −4.37983264279784316451336793887, −3.09428415314027232201116418594, −2.25288487243295406711082711960, −0.14844604267786918251094262128, 1.73133344228739641067116608804, 3.78628784388494463913476118255, 3.93589391652996748306899650805, 5.09947955730652492697323486017, 6.24982682638733796889779509273, 7.19324068356944653037168317602, 8.213806234456429725807504407086, 8.797963965303228813876269835145, 9.898511690464895932606735839967, 10.46465667009714537382434523075

Graph of the ZZ-function along the critical line