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Properties

Label 2-7616-1.1-c1-0-21
Degree 22
Conductor 76167616
Sign 11
Analytic cond. 60.814060.8140
Root an. cond. 7.798337.79833
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  − 1.61·3-s − 0.618·5-s + 7-s − 0.381·9-s − 0.763·11-s − 1.23·13-s + 1.00·15-s + 17-s − 8.47·19-s − 1.61·21-s + 7.70·23-s − 4.61·25-s + 5.47·27-s + 5.70·29-s − 6.32·31-s + 1.23·33-s − 0.618·35-s − 0.472·37-s + 2.00·39-s − 0.0901·41-s − 12.0·43-s + 0.236·45-s + 8.47·47-s + 49-s − 1.61·51-s − 10.7·53-s + 0.472·55-s + ⋯
L(s)  = 1  − 0.934·3-s − 0.276·5-s + 0.377·7-s − 0.127·9-s − 0.230·11-s − 0.342·13-s + 0.258·15-s + 0.242·17-s − 1.94·19-s − 0.353·21-s + 1.60·23-s − 0.923·25-s + 1.05·27-s + 1.05·29-s − 1.13·31-s + 0.215·33-s − 0.104·35-s − 0.0776·37-s + 0.320·39-s − 0.0140·41-s − 1.84·43-s + 0.0351·45-s + 1.23·47-s + 0.142·49-s − 0.226·51-s − 1.48·53-s + 0.0636·55-s + ⋯

Functional equation

Λ(s)=(7616s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7616s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76167616    =    26⋅7⋅172^{6} \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 60.814060.8140
Root analytic conductor: 7.798337.79833
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7616, ( :1/2), 1)(2,\ 7616,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.75103579980.7510357998
L(12)L(\frac12) ≈\approx 0.75103579980.7510357998
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
7 1−T 1 - T
17 1−T 1 - T
good3 1+1.61T+3T2 1 + 1.61T + 3T^{2}
5 1+0.618T+5T2 1 + 0.618T + 5T^{2}
11 1+0.763T+11T2 1 + 0.763T + 11T^{2}
13 1+1.23T+13T2 1 + 1.23T + 13T^{2}
19 1+8.47T+19T2 1 + 8.47T + 19T^{2}
23 1−7.70T+23T2 1 - 7.70T + 23T^{2}
29 1−5.70T+29T2 1 - 5.70T + 29T^{2}
31 1+6.32T+31T2 1 + 6.32T + 31T^{2}
37 1+0.472T+37T2 1 + 0.472T + 37T^{2}
41 1+0.0901T+41T2 1 + 0.0901T + 41T^{2}
43 1+12.0T+43T2 1 + 12.0T + 43T^{2}
47 1−8.47T+47T2 1 - 8.47T + 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−7.32T+61T2 1 - 7.32T + 61T^{2}
67 1+13.0T+67T2 1 + 13.0T + 67T^{2}
71 1+10.9T+71T2 1 + 10.9T + 71T^{2}
73 1−7.14T+73T2 1 - 7.14T + 73T^{2}
79 1+2.94T+79T2 1 + 2.94T + 79T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1−15.0T+97T2 1 - 15.0T + 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

−7.85321447766400466279884035026, −7.07108608951389518664019393260, −6.43328215411556502267113282878, −5.80776637951241252556581362967, −4.98042881469667387876819810878, −4.61045239395297084914147401505, −3.59647017718094309831269773891, −2.66819410727380007603645683899, −1.69010823860617216346377122733, −0.44895121116020088799937542157, 0.44895121116020088799937542157, 1.69010823860617216346377122733, 2.66819410727380007603645683899, 3.59647017718094309831269773891, 4.61045239395297084914147401505, 4.98042881469667387876819810878, 5.80776637951241252556581362967, 6.43328215411556502267113282878, 7.07108608951389518664019393260, 7.85321447766400466279884035026

Graph of the ZZ-function along the critical line