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Properties

Label 2-234-1.1-c1-0-1
Degree 22
Conductor 234234
Sign 11
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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 + 4-s − 2·5-s + 4·7-s + 8-s − 2·10-s + 4·11-s + 13-s + 4·14-s + 16-s − 2·17-s − 8·19-s − 2·20-s + 4·22-s − 25-s + 26-s + 4·28-s − 6·29-s − 4·31-s + 32-s − 2·34-s − 8·35-s − 2·37-s − 8·38-s − 2·40-s + 10·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.35·35-s − 0.328·37-s − 1.29·38-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 1)(2,\ 234,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8375079961.837507996
L(12)L(\frac12) ≈\approx 1.8375079961.837507996
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
3 1 1
13 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1−8T+pT2 1 - 8 T + 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+12T+pT2 1 + 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−12.06877268824115483606570314322, −11.29067241425549122912863548992, −10.82564350612112779643519160383, −9.025474627597816818050752077333, −8.129127522279784382432147914457, −7.16010307294570363491473287814, −5.93418826319229104786149140241, −4.49666099560716857332474018676, −3.92289433795456230197409104475, −1.87879273921165196191180493374, 1.87879273921165196191180493374, 3.92289433795456230197409104475, 4.49666099560716857332474018676, 5.93418826319229104786149140241, 7.16010307294570363491473287814, 8.129127522279784382432147914457, 9.025474627597816818050752077333, 10.82564350612112779643519160383, 11.29067241425549122912863548992, 12.06877268824115483606570314322

Graph of the ZZ-function along the critical line