↑

Properties

Label 2-26-13.9-c7-0-1
Degree 22
Conductor 2626
Sign 0.471−0.881i0.471 - 0.881i
Analytic cond. 8.122018.12201
Root an. cond. 2.849912.84991
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (−15.9 + 27.6i)3-s + (−31.9 − 55.4i)4-s − 54.4·5-s + (127. + 220. i)6-s + (556. + 963. i)7-s − 511.·8-s + (585. + 1.01e3i)9-s + (−217. + 377. i)10-s + (−3.56e3 + 6.17e3i)11-s + 2.03e3·12-s + (7.56e3 − 2.34e3i)13-s + 8.90e3·14-s + (867. − 1.50e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (1.02e4 + 1.77e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.340 + 0.590i)3-s + (−0.249 − 0.433i)4-s − 0.194·5-s + (0.240 + 0.417i)6-s + (0.613 + 1.06i)7-s − 0.353·8-s + (0.267 + 0.463i)9-s + (−0.0688 + 0.119i)10-s + (−0.807 + 1.39i)11-s + 0.340·12-s + (0.955 − 0.296i)13-s + 0.867·14-s + (0.0663 − 0.114i)15-s + (−0.125 + 0.216i)16-s + (0.506 + 0.876i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((0.471−0.881i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+7/2) L(s)=((0.471−0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 0.471−0.881i0.471 - 0.881i
Analytic conductor: 8.122018.12201
Root analytic conductor: 2.849912.84991
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ26(9,⋅)\chi_{26} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :7/2), 0.471−0.881i)(2,\ 26,\ (\ :7/2),\ 0.471 - 0.881i)

Particular Values

L(4)L(4) ≈\approx 1.32876+0.796255i1.32876 + 0.796255i
L(12)L(\frac12) ≈\approx 1.32876+0.796255i1.32876 + 0.796255i
L(92)L(\frac{9}{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+(−4+6.92i)T 1 + (-4 + 6.92i)T
13 1+(−7.56e3+2.34e3i)T 1 + (-7.56e3 + 2.34e3i)T
good3 1+(15.9−27.6i)T+(−1.09e3−1.89e3i)T2 1 + (15.9 - 27.6i)T + (-1.09e3 - 1.89e3i)T^{2}
5 1+54.4T+7.81e4T2 1 + 54.4T + 7.81e4T^{2}
7 1+(−556.−963.i)T+(−4.11e5+7.13e5i)T2 1 + (-556. - 963. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+(3.56e3−6.17e3i)T+(−9.74e6−1.68e7i)T2 1 + (3.56e3 - 6.17e3i)T + (-9.74e6 - 1.68e7i)T^{2}
17 1+(−1.02e4−1.77e4i)T+(−2.05e8+3.55e8i)T2 1 + (-1.02e4 - 1.77e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(1.40e4+2.43e4i)T+(−4.46e8+7.74e8i)T2 1 + (1.40e4 + 2.43e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+(1.69e4−2.93e4i)T+(−1.70e9−2.94e9i)T2 1 + (1.69e4 - 2.93e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−8.77e4+1.51e5i)T+(−8.62e9−1.49e10i)T2 1 + (-8.77e4 + 1.51e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1−1.56e4T+2.75e10T2 1 - 1.56e4T + 2.75e10T^{2}
37 1+(998.−1.72e3i)T+(−4.74e10−8.22e10i)T2 1 + (998. - 1.72e3i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+(1.11e5−1.93e5i)T+(−9.73e10−1.68e11i)T2 1 + (1.11e5 - 1.93e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(−2.68e5−4.65e5i)T+(−1.35e11+2.35e11i)T2 1 + (-2.68e5 - 4.65e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1−5.42e5T+5.06e11T2 1 - 5.42e5T + 5.06e11T^{2}
53 1−1.85e6T+1.17e12T2 1 - 1.85e6T + 1.17e12T^{2}
59 1+(−6.65e5−1.15e6i)T+(−1.24e12+2.15e12i)T2 1 + (-6.65e5 - 1.15e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.43e6+2.48e6i)T+(−1.57e12+2.72e12i)T2 1 + (1.43e6 + 2.48e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−1.41e6+2.45e6i)T+(−3.03e12−5.24e12i)T2 1 + (-1.41e6 + 2.45e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+(−8.00e5−1.38e6i)T+(−4.54e12+7.87e12i)T2 1 + (-8.00e5 - 1.38e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1−1.39e6T+1.10e13T2 1 - 1.39e6T + 1.10e13T^{2}
79 1+2.33e6T+1.92e13T2 1 + 2.33e6T + 1.92e13T^{2}
83 1−2.37e6T+2.71e13T2 1 - 2.37e6T + 2.71e13T^{2}
89 1+(3.52e6−6.10e6i)T+(−2.21e13−3.83e13i)T2 1 + (3.52e6 - 6.10e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+(4.25e6+7.37e6i)T+(−4.03e13+6.99e13i)T2 1 + (4.25e6 + 7.37e6i)T + (-4.03e13 + 6.99e13i)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

−15.61343049491279625869924254534, −15.17594893357944776738699853944, −13.35578815406339104687970231013, −12.14591695924005880568788672485, −10.94936682865697185767004653392, −9.832907861596866597401421165562, −8.082845610661017241180821525442, −5.61392653010655724868642819762, −4.36270168228763664302675118250, −2.10824556057425073623084577750, 0.77624088404093377737204112699, 3.83752993980241522668129601175, 5.78443546173782474486460611623, 7.20767196296221803248795093025, 8.412601132366499250411333218446, 10.63579854752849494014744812124, 11.96292496787501535742903930285, 13.41791173636955681392993540769, 14.17034724646112985847384806431, 15.84032261032378979759255502373

Graph of the ZZ-function along the critical line