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Properties

Label 2-950-1.1-c1-0-18
Degree 22
Conductor 950950
Sign 11
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s − 3·6-s + 5·7-s − 8-s + 6·9-s − 4·11-s + 3·12-s + 13-s − 5·14-s + 16-s + 3·17-s − 6·18-s + 19-s + 15·21-s + 4·22-s − 7·23-s − 3·24-s − 26-s + 9·27-s + 5·28-s − 3·29-s − 2·31-s − 32-s − 12·33-s − 3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 1.88·7-s − 0.353·8-s + 2·9-s − 1.20·11-s + 0.866·12-s + 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s + 3.27·21-s + 0.852·22-s − 1.45·23-s − 0.612·24-s − 0.196·26-s + 1.73·27-s + 0.944·28-s − 0.557·29-s − 0.359·31-s − 0.176·32-s − 2.08·33-s − 0.514·34-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 1)(2,\ 950,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4396504052.439650405
L(12)L(\frac12) ≈\approx 2.4396504052.439650405
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+T 1 + T
5 1 1
19 1−T 1 - T
good3 1−pT+pT2 1 - p T + p T^{2}
7 1−5T+pT2 1 - 5 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1+7T+pT2 1 + 7 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−13T+pT2 1 - 13 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−10T+pT2 1 - 10 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p 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

−9.976858889227114824891335123219, −8.920485942669728815604036529685, −8.301058617244315182063927960266, −7.80203381755481970407982696068, −7.37288434104770513673343989750, −5.66402186780550341225871252054, −4.58979652776477679003498012022, −3.44171251094378542463371483956, −2.28599476561718030727276453764, −1.57691270891887807421558967069, 1.57691270891887807421558967069, 2.28599476561718030727276453764, 3.44171251094378542463371483956, 4.58979652776477679003498012022, 5.66402186780550341225871252054, 7.37288434104770513673343989750, 7.80203381755481970407982696068, 8.301058617244315182063927960266, 8.920485942669728815604036529685, 9.976858889227114824891335123219

Graph of the ZZ-function along the critical line