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Properties

Label 2-4925-1.1-c1-0-0
Degree 22
Conductor 49254925
Sign 11
Analytic cond. 39.326339.3263
Root an. cond. 6.271076.27107
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  + 0.411·2-s + 0.231·3-s − 1.83·4-s + 0.0951·6-s − 3.56·7-s − 1.57·8-s − 2.94·9-s − 4.00·11-s − 0.423·12-s − 3.83·13-s − 1.46·14-s + 3.01·16-s − 0.709·17-s − 1.21·18-s − 6.25·19-s − 0.824·21-s − 1.64·22-s − 1.59·23-s − 0.364·24-s − 1.57·26-s − 1.37·27-s + 6.53·28-s + 2.22·29-s + 1.31·31-s + 4.39·32-s − 0.925·33-s − 0.291·34-s + ⋯
L(s)  = 1  + 0.291·2-s + 0.133·3-s − 0.915·4-s + 0.0388·6-s − 1.34·7-s − 0.557·8-s − 0.982·9-s − 1.20·11-s − 0.122·12-s − 1.06·13-s − 0.392·14-s + 0.753·16-s − 0.172·17-s − 0.285·18-s − 1.43·19-s − 0.179·21-s − 0.351·22-s − 0.332·23-s − 0.0743·24-s − 0.309·26-s − 0.264·27-s + 1.23·28-s + 0.412·29-s + 0.235·31-s + 0.776·32-s − 0.161·33-s − 0.0500·34-s + ⋯

Functional equation

Λ(s)=(4925s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4925s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49254925    =    52⋅1975^{2} \cdot 197
Sign: 11
Analytic conductor: 39.326339.3263
Root analytic conductor: 6.271076.27107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4925, ( :1/2), 1)(2,\ 4925,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.0043208078440.004320807844
L(12)L(\frac12) ≈\approx 0.0043208078440.004320807844
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)
bad5 1 1
197 1+T 1 + T
good2 1−0.411T+2T2 1 - 0.411T + 2T^{2}
3 1−0.231T+3T2 1 - 0.231T + 3T^{2}
7 1+3.56T+7T2 1 + 3.56T + 7T^{2}
11 1+4.00T+11T2 1 + 4.00T + 11T^{2}
13 1+3.83T+13T2 1 + 3.83T + 13T^{2}
17 1+0.709T+17T2 1 + 0.709T + 17T^{2}
19 1+6.25T+19T2 1 + 6.25T + 19T^{2}
23 1+1.59T+23T2 1 + 1.59T + 23T^{2}
29 1−2.22T+29T2 1 - 2.22T + 29T^{2}
31 1−1.31T+31T2 1 - 1.31T + 31T^{2}
37 1+5.52T+37T2 1 + 5.52T + 37T^{2}
41 1+10.8T+41T2 1 + 10.8T + 41T^{2}
43 1−3.23T+43T2 1 - 3.23T + 43T^{2}
47 1+7.97T+47T2 1 + 7.97T + 47T^{2}
53 1+9.53T+53T2 1 + 9.53T + 53T^{2}
59 1+10.4T+59T2 1 + 10.4T + 59T^{2}
61 1+11.5T+61T2 1 + 11.5T + 61T^{2}
67 1−5.42T+67T2 1 - 5.42T + 67T^{2}
71 1−7.54T+71T2 1 - 7.54T + 71T^{2}
73 1−0.717T+73T2 1 - 0.717T + 73T^{2}
79 1+0.425T+79T2 1 + 0.425T + 79T^{2}
83 1−12.5T+83T2 1 - 12.5T + 83T^{2}
89 1−15.8T+89T2 1 - 15.8T + 89T^{2}
97 1−16.8T+97T2 1 - 16.8T + 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

−8.249082798980141219357933699998, −7.74824668387358754489691891587, −6.53008270739412503288411733592, −6.16580938178917925047873336210, −5.12666836931183465946116922213, −4.76250935045636058037798924688, −3.57638853613800261660574586756, −3.03629202119177778667743602958, −2.22317449680797359443373612053, −0.03131565452827996904895270360, 0.03131565452827996904895270360, 2.22317449680797359443373612053, 3.03629202119177778667743602958, 3.57638853613800261660574586756, 4.76250935045636058037798924688, 5.12666836931183465946116922213, 6.16580938178917925047873336210, 6.53008270739412503288411733592, 7.74824668387358754489691891587, 8.249082798980141219357933699998

Graph of the ZZ-function along the critical line