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Properties

Label 2-296-1.1-c1-0-6
Degree 22
Conductor 296296
Sign 11
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + 2.47·3-s + 1.47·5-s + 2.64·7-s + 3.11·9-s − 6.34·11-s − 5.34·13-s + 3.64·15-s − 0.715·17-s + 3.28·19-s + 6.53·21-s + 0.885·23-s − 2.83·25-s + 0.284·27-s − 0.885·29-s + 7.47·31-s − 15.6·33-s + 3.89·35-s − 37-s − 13.2·39-s − 7.75·41-s + 10.1·43-s + 4.58·45-s + 11.8·47-s − 0.0194·49-s − 1.77·51-s + 4.15·53-s − 9.34·55-s + ⋯
L(s)  = 1  + 1.42·3-s + 0.658·5-s + 0.998·7-s + 1.03·9-s − 1.91·11-s − 1.48·13-s + 0.940·15-s − 0.173·17-s + 0.753·19-s + 1.42·21-s + 0.184·23-s − 0.566·25-s + 0.0546·27-s − 0.164·29-s + 1.34·31-s − 2.73·33-s + 0.657·35-s − 0.164·37-s − 2.11·39-s − 1.21·41-s + 1.54·43-s + 0.683·45-s + 1.72·47-s − 0.00277·49-s − 0.247·51-s + 0.570·53-s − 1.26·55-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 11
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 1)(2,\ 296,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0931794952.093179495
L(12)L(\frac12) ≈\approx 2.0931794952.093179495
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
37 1+T 1 + T
good3 1−2.47T+3T2 1 - 2.47T + 3T^{2}
5 1−1.47T+5T2 1 - 1.47T + 5T^{2}
7 1−2.64T+7T2 1 - 2.64T + 7T^{2}
11 1+6.34T+11T2 1 + 6.34T + 11T^{2}
13 1+5.34T+13T2 1 + 5.34T + 13T^{2}
17 1+0.715T+17T2 1 + 0.715T + 17T^{2}
19 1−3.28T+19T2 1 - 3.28T + 19T^{2}
23 1−0.885T+23T2 1 - 0.885T + 23T^{2}
29 1+0.885T+29T2 1 + 0.885T + 29T^{2}
31 1−7.47T+31T2 1 - 7.47T + 31T^{2}
41 1+7.75T+41T2 1 + 7.75T + 41T^{2}
43 1−10.1T+43T2 1 - 10.1T + 43T^{2}
47 1−11.8T+47T2 1 - 11.8T + 47T^{2}
53 1−4.15T+53T2 1 - 4.15T + 53T^{2}
59 1+12.1T+59T2 1 + 12.1T + 59T^{2}
61 1−3.81T+61T2 1 - 3.81T + 61T^{2}
67 1+5.32T+67T2 1 + 5.32T + 67T^{2}
71 1−2.87T+71T2 1 - 2.87T + 71T^{2}
73 1+8.34T+73T2 1 + 8.34T + 73T^{2}
79 1+1.83T+79T2 1 + 1.83T + 79T^{2}
83 1−2.15T+83T2 1 - 2.15T + 83T^{2}
89 1+1.89T+89T2 1 + 1.89T + 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

−11.85605980907732818955785938353, −10.50667674794395328904484860416, −9.844634532495756245168498798666, −8.863899326492502878201953865086, −7.83502155766063885308161734339, −7.46566984707687402433081972410, −5.54008877409685999977956279070, −4.63674996738275428236758451517, −2.84942306405856913627278075228, −2.14416178081521092823242555565, 2.14416178081521092823242555565, 2.84942306405856913627278075228, 4.63674996738275428236758451517, 5.54008877409685999977956279070, 7.46566984707687402433081972410, 7.83502155766063885308161734339, 8.863899326492502878201953865086, 9.844634532495756245168498798666, 10.50667674794395328904484860416, 11.85605980907732818955785938353

Graph of the ZZ-function along the critical line