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Properties

Label 2-34-1.1-c1-0-0
Degree 22
Conductor 3434
Sign 11
Analytic cond. 0.2714910.271491
Root an. cond. 0.5210480.521048
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 − 2·3-s + 4-s − 2·6-s − 4·7-s + 8-s + 9-s + 6·11-s − 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s + 18-s − 4·19-s + 8·21-s + 6·22-s − 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s − 4·31-s + 32-s − 12·33-s − 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s − 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s − 0.718·31-s + 0.176·32-s − 2.08·33-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(34s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(34s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3434    =    2⋅172 \cdot 17
Sign: 11
Analytic conductor: 0.2714910.271491
Root analytic conductor: 0.5210480.521048
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 34, ( :1/2), 1)(2,\ 34,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.74927722100.7492772210
L(12)L(\frac12) ≈\approx 0.74927722100.7492772210
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
17 1+T 1 + T
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−16.63179484427491344136926072682, −15.73128059955905922020314753971, −14.21978494242446158082804784774, −12.85956463289156748627394436952, −11.95834417367800404279447796077, −10.84894668716717132123408330664, −9.286071988264369821451329617999, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −3.90229547122613769308947697962, 3.90229547122613769308947697962, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 9.286071988264369821451329617999, 10.84894668716717132123408330664, 11.95834417367800404279447796077, 12.85956463289156748627394436952, 14.21978494242446158082804784774, 15.73128059955905922020314753971, 16.63179484427491344136926072682

Graph of the ZZ-function along the critical line