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Properties

Label 2-2e10-16.13-c1-0-23
Degree 22
Conductor 10241024
Sign −0.382+0.923i-0.382 + 0.923i
Analytic cond. 8.176688.17668
Root an. cond. 2.859482.85948
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  + (1.41 − 1.41i)5-s − 3i·9-s + (−4.24 − 4.24i)13-s − 2·17-s + 0.999i·25-s + (−7.07 − 7.07i)29-s + (1.41 − 1.41i)37-s + 10i·41-s + (−4.24 − 4.24i)45-s + 7·49-s + (9.89 − 9.89i)53-s + (−7.07 − 7.07i)61-s − 12·65-s − 6i·73-s − 9·81-s + ⋯
L(s)  = 1  + (0.632 − 0.632i)5-s − i·9-s + (−1.17 − 1.17i)13-s − 0.485·17-s + 0.199i·25-s + (−1.31 − 1.31i)29-s + (0.232 − 0.232i)37-s + 1.56i·41-s + (−0.632 − 0.632i)45-s + 49-s + (1.35 − 1.35i)53-s + (−0.905 − 0.905i)61-s − 1.48·65-s − 0.702i·73-s − 81-s + ⋯

Functional equation

Λ(s)=(1024s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1024s/2 ΓC(s+1/2) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10241024    =    2102^{10}
Sign: −0.382+0.923i-0.382 + 0.923i
Analytic conductor: 8.176688.17668
Root analytic conductor: 2.859482.85948
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1024(257,⋅)\chi_{1024} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1024, ( :1/2), −0.382+0.923i)(2,\ 1024,\ (\ :1/2),\ -0.382 + 0.923i)

Particular Values

L(1)L(1) ≈\approx 1.3110287771.311028777
L(12)L(\frac12) ≈\approx 1.3110287771.311028777
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
good3 1+3iT2 1 + 3iT^{2}
5 1+(−1.41+1.41i)T−5iT2 1 + (-1.41 + 1.41i)T - 5iT^{2}
7 1−7T2 1 - 7T^{2}
11 1−11iT2 1 - 11iT^{2}
13 1+(4.24+4.24i)T+13iT2 1 + (4.24 + 4.24i)T + 13iT^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+19iT2 1 + 19iT^{2}
23 1−23T2 1 - 23T^{2}
29 1+(7.07+7.07i)T+29iT2 1 + (7.07 + 7.07i)T + 29iT^{2}
31 1+31T2 1 + 31T^{2}
37 1+(−1.41+1.41i)T−37iT2 1 + (-1.41 + 1.41i)T - 37iT^{2}
41 1−10iT−41T2 1 - 10iT - 41T^{2}
43 1−43iT2 1 - 43iT^{2}
47 1+47T2 1 + 47T^{2}
53 1+(−9.89+9.89i)T−53iT2 1 + (-9.89 + 9.89i)T - 53iT^{2}
59 1−59iT2 1 - 59iT^{2}
61 1+(7.07+7.07i)T+61iT2 1 + (7.07 + 7.07i)T + 61iT^{2}
67 1+67iT2 1 + 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1+10iT−89T2 1 + 10iT - 89T^{2}
97 1−18T+97T2 1 - 18T + 97T^{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

−9.620283036764787944307423816710, −9.064602094731346530248000317992, −8.037174266388778325042618447529, −7.20843442642607362112930182312, −6.11847789407009324111669185490, −5.44051043365300152293167439512, −4.48202280828293009755043746083, −3.29094604743136645679953673768, −2.09309480223514768698021602910, −0.56167569235541526640622874703, 1.93156024107026740741843752164, 2.60007298966074515473555153780, 4.08713117179827191216354712982, 5.04205152256619503955879385253, 5.91652349504064375793631567283, 7.05859567686355949057528065175, 7.38519512176941067332095039794, 8.718483851383060738656172169502, 9.377663218754535898129479680363, 10.33337060286031108206974089336

Graph of the ZZ-function along the critical line