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Properties

Label 2-1785-1.1-c1-0-53
Degree 22
Conductor 17851785
Sign 11
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.59·2-s + 3-s + 4.74·4-s − 5-s + 2.59·6-s + 7-s + 7.11·8-s + 9-s − 2.59·10-s − 1.74·11-s + 4.74·12-s + 5.74·13-s + 2.59·14-s − 15-s + 8.99·16-s + 17-s + 2.59·18-s − 0.514·19-s − 4.74·20-s + 21-s − 4.52·22-s − 8.26·23-s + 7.11·24-s + 25-s + 14.9·26-s + 27-s + 4.74·28-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.577·3-s + 2.37·4-s − 0.447·5-s + 1.05·6-s + 0.377·7-s + 2.51·8-s + 0.333·9-s − 0.821·10-s − 0.524·11-s + 1.36·12-s + 1.59·13-s + 0.693·14-s − 0.258·15-s + 2.24·16-s + 0.242·17-s + 0.611·18-s − 0.117·19-s − 1.06·20-s + 0.218·21-s − 0.963·22-s − 1.72·23-s + 1.45·24-s + 0.200·25-s + 2.92·26-s + 0.192·27-s + 0.895·28-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), 1)(2,\ 1785,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.4838000986.483800098
L(12)L(\frac12) ≈\approx 6.4838000986.483800098
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)
bad3 1−T 1 - T
5 1+T 1 + T
7 1−T 1 - T
17 1−T 1 - T
good2 1−2.59T+2T2 1 - 2.59T + 2T^{2}
11 1+1.74T+11T2 1 + 1.74T + 11T^{2}
13 1−5.74T+13T2 1 - 5.74T + 13T^{2}
19 1+0.514T+19T2 1 + 0.514T + 19T^{2}
23 1+8.26T+23T2 1 + 8.26T + 23T^{2}
29 1+4.52T+29T2 1 + 4.52T + 29T^{2}
31 1+1.74T+31T2 1 + 1.74T + 31T^{2}
37 1−2.77T+37T2 1 - 2.77T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−4.41T+43T2 1 - 4.41T + 43T^{2}
47 1−4.51T+47T2 1 - 4.51T + 47T^{2}
53 1+12.2T+53T2 1 + 12.2T + 53T^{2}
59 1−2.51T+59T2 1 - 2.51T + 59T^{2}
61 1−5.96T+61T2 1 - 5.96T + 61T^{2}
67 1+7.31T+67T2 1 + 7.31T + 67T^{2}
71 1−10.7T+71T2 1 - 10.7T + 71T^{2}
73 1+7.74T+73T2 1 + 7.74T + 73T^{2}
79 1−11.0T+79T2 1 - 11.0T + 79T^{2}
83 1−7.99T+83T2 1 - 7.99T + 83T^{2}
89 1−8.77T+89T2 1 - 8.77T + 89T^{2}
97 1+2.70T+97T2 1 + 2.70T + 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

−9.214604665630104221778810203124, −8.069631842531778821122490581293, −7.72945316178797549090564108505, −6.60133766851824750728756065249, −5.91001204956921388817028342599, −5.11469510534396608698352926129, −4.00437861004790193435166845705, −3.75027385637544515760897095756, −2.65140239244998467924407652444, −1.64041601859189390881405630255, 1.64041601859189390881405630255, 2.65140239244998467924407652444, 3.75027385637544515760897095756, 4.00437861004790193435166845705, 5.11469510534396608698352926129, 5.91001204956921388817028342599, 6.60133766851824750728756065249, 7.72945316178797549090564108505, 8.069631842531778821122490581293, 9.214604665630104221778810203124

Graph of the ZZ-function along the critical line