↑

Properties

Label 2-29e2-29.9-c1-0-34
Degree 22
Conductor 841841
Sign 0.974+0.225i0.974 + 0.225i
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
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  + (0.268 + 0.556i)2-s + (−0.483 + 0.385i)3-s + (1.00 − 1.26i)4-s + (3.47 − 1.67i)5-s + (−0.344 − 0.165i)6-s + (−1.39 − 1.74i)7-s + (2.18 + 0.497i)8-s + (−0.582 + 2.55i)9-s + (1.86 + 1.48i)10-s + (1.34 − 0.307i)11-s + i·12-s + (−0.0525 − 0.230i)13-s + (0.599 − 1.24i)14-s + (−1.03 + 2.14i)15-s + (−0.412 − 1.80i)16-s + 4.38i·17-s + ⋯
L(s)  = 1  + (0.189 + 0.393i)2-s + (−0.278 + 0.222i)3-s + (0.504 − 0.632i)4-s + (1.55 − 0.747i)5-s + (−0.140 − 0.0676i)6-s + (−0.526 − 0.660i)7-s + (0.770 + 0.175i)8-s + (−0.194 + 0.850i)9-s + (0.588 + 0.469i)10-s + (0.406 − 0.0927i)11-s + 0.288i·12-s + (−0.0145 − 0.0638i)13-s + (0.160 − 0.332i)14-s + (−0.266 + 0.554i)15-s + (−0.103 − 0.451i)16-s + 1.06i·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((0.974+0.225i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((0.974+0.225i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: 0.974+0.225i0.974 + 0.225i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(270,⋅)\chi_{841} (270, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), 0.974+0.225i)(2,\ 841,\ (\ :1/2),\ 0.974 + 0.225i)

Particular Values

L(1)L(1) ≈\approx 2.29435−0.261550i2.29435 - 0.261550i
L(12)L(\frac12) ≈\approx 2.29435−0.261550i2.29435 - 0.261550i
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)
bad29 1 1
good2 1+(−0.268−0.556i)T+(−1.24+1.56i)T2 1 + (-0.268 - 0.556i)T + (-1.24 + 1.56i)T^{2}
3 1+(0.483−0.385i)T+(0.667−2.92i)T2 1 + (0.483 - 0.385i)T + (0.667 - 2.92i)T^{2}
5 1+(−3.47+1.67i)T+(3.11−3.90i)T2 1 + (-3.47 + 1.67i)T + (3.11 - 3.90i)T^{2}
7 1+(1.39+1.74i)T+(−1.55+6.82i)T2 1 + (1.39 + 1.74i)T + (-1.55 + 6.82i)T^{2}
11 1+(−1.34+0.307i)T+(9.91−4.77i)T2 1 + (-1.34 + 0.307i)T + (9.91 - 4.77i)T^{2}
13 1+(0.0525+0.230i)T+(−11.7+5.64i)T2 1 + (0.0525 + 0.230i)T + (-11.7 + 5.64i)T^{2}
17 1−4.38iT−17T2 1 - 4.38iT - 17T^{2}
19 1+(−3.79−3.02i)T+(4.22+18.5i)T2 1 + (-3.79 - 3.02i)T + (4.22 + 18.5i)T^{2}
23 1+(−1.11−0.536i)T+(14.3+17.9i)T2 1 + (-1.11 - 0.536i)T + (14.3 + 17.9i)T^{2}
31 1+(4.37+9.09i)T+(−19.3+24.2i)T2 1 + (4.37 + 9.09i)T + (-19.3 + 24.2i)T^{2}
37 1+(4.59+1.04i)T+(33.3+16.0i)T2 1 + (4.59 + 1.04i)T + (33.3 + 16.0i)T^{2}
41 1−3.85iT−41T2 1 - 3.85iT - 41T^{2}
43 1+(−3.13+6.51i)T+(−26.8−33.6i)T2 1 + (-3.13 + 6.51i)T + (-26.8 - 33.6i)T^{2}
47 1+(6.82−1.55i)T+(42.3−20.3i)T2 1 + (6.82 - 1.55i)T + (42.3 - 20.3i)T^{2}
53 1+(−1.80+0.867i)T+(33.0−41.4i)T2 1 + (-1.80 + 0.867i)T + (33.0 - 41.4i)T^{2}
59 1−6.09T+59T2 1 - 6.09T + 59T^{2}
61 1+(0.483−0.385i)T+(13.5−59.4i)T2 1 + (0.483 - 0.385i)T + (13.5 - 59.4i)T^{2}
67 1+(0.339−1.48i)T+(−60.3−29.0i)T2 1 + (0.339 - 1.48i)T + (-60.3 - 29.0i)T^{2}
71 1+(−2.33−10.2i)T+(−63.9+30.8i)T2 1 + (-2.33 - 10.2i)T + (-63.9 + 30.8i)T^{2}
73 1+(5.94−12.3i)T+(−45.5−57.0i)T2 1 + (5.94 - 12.3i)T + (-45.5 - 57.0i)T^{2}
79 1+(5.93+1.35i)T+(71.1+34.2i)T2 1 + (5.93 + 1.35i)T + (71.1 + 34.2i)T^{2}
83 1+(6.20−7.77i)T+(−18.4−80.9i)T2 1 + (6.20 - 7.77i)T + (-18.4 - 80.9i)T^{2}
89 1+(−2.04−4.24i)T+(−55.4+69.5i)T2 1 + (-2.04 - 4.24i)T + (-55.4 + 69.5i)T^{2}
97 1+(−2.78−2.22i)T+(21.5+94.5i)T2 1 + (-2.78 - 2.22i)T + (21.5 + 94.5i)T^{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

−10.03421260845797390667776207487, −9.695562789867447831147096638734, −8.517814706729050892492237645071, −7.41958725448610246561045206811, −6.42746754220080940431438546614, −5.67191427353974833614889440510, −5.27562976120342545025929740840, −4.02807800356887535552407741813, −2.24948435261610742183453717790, −1.28630566606136308687533845422, 1.60578060547623550960346853358, 2.81093300795874305160856761746, 3.28019624937593781073364121165, 5.05787770289355342679269199933, 6.02182471024711003030988350320, 6.76357733486929382663681140612, 7.22585570419334655036348968948, 8.932150416736550701002117879392, 9.381434940019659459543516312617, 10.27749940651262789657600911089

Graph of the ZZ-function along the critical line