↑

Properties

Label 2-13e2-169.17-c1-0-1
Degree 22
Conductor 169169
Sign −0.0987−0.995i-0.0987 - 0.995i
Analytic cond. 1.349471.34947
Root an. cond. 1.161661.16166
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  + (−1.98 + 0.404i)2-s + (−0.359 − 0.0584i)3-s + (1.92 − 0.819i)4-s + (0.199 + 0.809i)5-s + (0.735 − 0.0296i)6-s + (0.512 + 0.243i)7-s + (−0.149 + 0.102i)8-s + (−2.71 − 0.908i)9-s + (−0.722 − 1.52i)10-s + (1.27 + 3.83i)11-s + (−0.738 + 0.182i)12-s + (2.71 + 2.37i)13-s + (−1.11 − 0.274i)14-s + (−0.0244 − 0.302i)15-s + (−2.64 + 2.74i)16-s + (−3.39 + 7.15i)17-s + ⋯
L(s)  = 1  + (−1.40 + 0.286i)2-s + (−0.207 − 0.0337i)3-s + (0.961 − 0.409i)4-s + (0.0892 + 0.362i)5-s + (0.300 − 0.0121i)6-s + (0.193 + 0.0919i)7-s + (−0.0527 + 0.0363i)8-s + (−0.906 − 0.302i)9-s + (−0.228 − 0.481i)10-s + (0.385 + 1.15i)11-s + (−0.213 + 0.0525i)12-s + (0.753 + 0.657i)13-s + (−0.297 − 0.0734i)14-s + (−0.00630 − 0.0781i)15-s + (−0.660 + 0.687i)16-s + (−0.822 + 1.73i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.0987−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((−0.0987−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.0987−0.995i-0.0987 - 0.995i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(17,⋅)\chi_{169} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), −0.0987−0.995i)(2,\ 169,\ (\ :1/2),\ -0.0987 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.319075+0.352290i0.319075 + 0.352290i
L(12)L(\frac12) ≈\approx 0.319075+0.352290i0.319075 + 0.352290i
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)
bad13 1+(−2.71−2.37i)T 1 + (-2.71 - 2.37i)T
good2 1+(1.98−0.404i)T+(1.83−0.783i)T2 1 + (1.98 - 0.404i)T + (1.83 - 0.783i)T^{2}
3 1+(0.359+0.0584i)T+(2.84+0.950i)T2 1 + (0.359 + 0.0584i)T + (2.84 + 0.950i)T^{2}
5 1+(−0.199−0.809i)T+(−4.42+2.32i)T2 1 + (-0.199 - 0.809i)T + (-4.42 + 2.32i)T^{2}
7 1+(−0.512−0.243i)T+(4.42+5.42i)T2 1 + (-0.512 - 0.243i)T + (4.42 + 5.42i)T^{2}
11 1+(−1.27−3.83i)T+(−8.79+6.60i)T2 1 + (-1.27 - 3.83i)T + (-8.79 + 6.60i)T^{2}
17 1+(3.39−7.15i)T+(−10.7−13.1i)T2 1 + (3.39 - 7.15i)T + (-10.7 - 13.1i)T^{2}
19 1+(−1.11+0.646i)T+(9.5−16.4i)T2 1 + (-1.11 + 0.646i)T + (9.5 - 16.4i)T^{2}
23 1+(3.79−6.57i)T+(−11.5−19.9i)T2 1 + (3.79 - 6.57i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.254+1.24i)T+(−26.6+11.3i)T2 1 + (0.254 + 1.24i)T + (-26.6 + 11.3i)T^{2}
31 1+(−1.29+2.46i)T+(−17.6−25.5i)T2 1 + (-1.29 + 2.46i)T + (-17.6 - 25.5i)T^{2}
37 1+(−0.238−0.377i)T+(−15.8+33.4i)T2 1 + (-0.238 - 0.377i)T + (-15.8 + 33.4i)T^{2}
41 1+(−0.704+4.33i)T+(−38.8−12.9i)T2 1 + (-0.704 + 4.33i)T + (-38.8 - 12.9i)T^{2}
43 1+(−6.85−4.33i)T+(18.4+38.8i)T2 1 + (-6.85 - 4.33i)T + (18.4 + 38.8i)T^{2}
47 1+(−2.97+0.361i)T+(45.6−11.2i)T2 1 + (-2.97 + 0.361i)T + (45.6 - 11.2i)T^{2}
53 1+(4.71+6.83i)T+(−18.7+49.5i)T2 1 + (4.71 + 6.83i)T + (-18.7 + 49.5i)T^{2}
59 1+(−3.83+3.68i)T+(2.37−58.9i)T2 1 + (-3.83 + 3.68i)T + (2.37 - 58.9i)T^{2}
61 1+(6.56+0.529i)T+(60.2+9.78i)T2 1 + (6.56 + 0.529i)T + (60.2 + 9.78i)T^{2}
67 1+(3.93−9.23i)T+(−46.4−48.3i)T2 1 + (3.93 - 9.23i)T + (-46.4 - 48.3i)T^{2}
71 1+(−10.8−8.81i)T+(14.2+69.5i)T2 1 + (-10.8 - 8.81i)T + (14.2 + 69.5i)T^{2}
73 1+(3.93−4.44i)T+(−8.79−72.4i)T2 1 + (3.93 - 4.44i)T + (-8.79 - 72.4i)T^{2}
79 1+(0.977+8.04i)T+(−76.7+18.9i)T2 1 + (0.977 + 8.04i)T + (-76.7 + 18.9i)T^{2}
83 1+(−5.62−2.13i)T+(62.1+55.0i)T2 1 + (-5.62 - 2.13i)T + (62.1 + 55.0i)T^{2}
89 1+(6.62+3.82i)T+(44.5+77.0i)T2 1 + (6.62 + 3.82i)T + (44.5 + 77.0i)T^{2}
97 1+(−9.36+2.71i)T+(81.9−51.8i)T2 1 + (-9.36 + 2.71i)T + (81.9 - 51.8i)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

−12.94912418187831044532137792900, −11.64878133612749930462367555836, −10.89377200767181369633523038199, −9.854761557608086393888896230694, −8.949551861463315254533998368212, −8.123024254627270289630916223944, −6.89962567596070211368179464955, −6.05028327354641079305105277076, −4.09622380000040505136747343512, −1.83031685246236075381973938398, 0.74216460973868753811302896239, 2.80671790630805568059989674683, 4.93214955910595350653913733995, 6.27722255396141827102257626034, 7.77101942651967379196368176479, 8.681785279800322952354124049196, 9.208976689419296337828785379780, 10.70157006076968059354669534841, 11.08121287135571253950049463177, 12.09640687828717514944093107484

Graph of the ZZ-function along the critical line