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Properties

Label 2-300-3.2-c8-0-13
Degree 22
Conductor 300300
Sign 11
Analytic cond. 122.213122.213
Root an. cond. 11.055011.0550
Motivic weight 88
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 81·3-s − 4.03e3·7-s + 6.56e3·9-s + 3.58e4·13-s − 2.58e5·19-s + 3.26e5·21-s − 5.31e5·27-s − 1.80e6·31-s − 5.03e5·37-s − 2.90e6·39-s − 3.49e6·43-s + 1.05e7·49-s + 2.09e7·57-s − 2.38e7·61-s − 2.64e7·63-s + 5.42e6·67-s − 1.61e7·73-s − 1.88e7·79-s + 4.30e7·81-s − 1.44e8·91-s + 1.46e8·93-s − 1.76e8·97-s − 4.44e7·103-s + 2.03e8·109-s + 4.07e7·111-s + 2.34e8·117-s + ⋯
L(s)  = 1  − 3-s − 1.68·7-s + 9-s + 1.25·13-s − 1.98·19-s + 1.68·21-s − 27-s − 1.95·31-s − 0.268·37-s − 1.25·39-s − 1.02·43-s + 1.82·49-s + 1.98·57-s − 1.72·61-s − 1.68·63-s + 0.269·67-s − 0.569·73-s − 0.484·79-s + 81-s − 2.10·91-s + 1.95·93-s − 1.99·97-s − 0.394·103-s + 1.43·109-s + 0.268·111-s + 1.25·117-s + ⋯

Functional equation

Λ(s)=(300s/2 ΓC(s) L(s)=( Λ(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
Λ(s)=(300s/2 ΓC(s+4) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 300300    =    22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 122.213122.213
Root analytic conductor: 11.055011.0550
Motivic weight: 88
Rational: yes
Arithmetic: yes
Character: χ300(101,⋅)\chi_{300} (101, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 300, ( :4), 1)(2,\ 300,\ (\ :4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.47182767030.4718276703
L(12)L(\frac12) ≈\approx 0.47182767030.4718276703
L(5)L(5) 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
3 1+p4T 1 + p^{4} T
5 1 1
good7 1+4034T+p8T2 1 + 4034 T + p^{8} T^{2}
11 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
13 1−35806T+p8T2 1 - 35806 T + p^{8} T^{2}
17 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
19 1+258526T+p8T2 1 + 258526 T + p^{8} T^{2}
23 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
29 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
31 1+1809406T+p8T2 1 + 1809406 T + p^{8} T^{2}
37 1+503522T+p8T2 1 + 503522 T + p^{8} T^{2}
41 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
43 1+3492194T+p8T2 1 + 3492194 T + p^{8} T^{2}
47 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
53 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
59 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
61 1+23826526T+p8T2 1 + 23826526 T + p^{8} T^{2}
67 1−5421406T+p8T2 1 - 5421406 T + p^{8} T^{2}
71 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
73 1+16169282T+p8T2 1 + 16169282 T + p^{8} T^{2}
79 1+18887038T+p8T2 1 + 18887038 T + p^{8} T^{2}
83 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
89 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
97 1+176908034T+p8T2 1 + 176908034 T + p^{8} T^{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

−10.51532472881295099328083754157, −9.539547682592394203517530895662, −8.599418640384385595667162935738, −7.07554235602067965365868065674, −6.34824307044109561116322102624, −5.72917899004978767422508595316, −4.24661101133806357365297148874, −3.36564351286681277656071627671, −1.75582856761493510062648144327, −0.32804838935866592474990223094, 0.32804838935866592474990223094, 1.75582856761493510062648144327, 3.36564351286681277656071627671, 4.24661101133806357365297148874, 5.72917899004978767422508595316, 6.34824307044109561116322102624, 7.07554235602067965365868065674, 8.599418640384385595667162935738, 9.539547682592394203517530895662, 10.51532472881295099328083754157

Graph of the ZZ-function along the critical line