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Properties

Label 2-51-51.50-c4-0-1
Degree 22
Conductor 5151
Sign 0.284−0.958i0.284 - 0.958i
Analytic cond. 5.271865.27186
Root an. cond. 2.296052.29605
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.45i·2-s + (1.27 + 8.90i)3-s − 39.5·4-s − 18.0·5-s + (66.3 − 9.51i)6-s + 41.9i·7-s + 175. i·8-s + (−77.7 + 22.7i)9-s + 134. i·10-s − 94.6·11-s + (−50.4 − 352. i)12-s + 128.·13-s + 312.·14-s + (−23.0 − 160. i)15-s + 674.·16-s + (−262. − 120. i)17-s + ⋯
L(s)  = 1  − 1.86i·2-s + (0.141 + 0.989i)3-s − 2.47·4-s − 0.721·5-s + (1.84 − 0.264i)6-s + 0.855i·7-s + 2.74i·8-s + (−0.959 + 0.280i)9-s + 1.34i·10-s − 0.782·11-s + (−0.350 − 2.44i)12-s + 0.761·13-s + 1.59·14-s + (−0.102 − 0.714i)15-s + 2.63·16-s + (−0.908 − 0.417i)17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((0.284−0.958i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+2) L(s)=((0.284−0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: 0.284−0.958i0.284 - 0.958i
Analytic conductor: 5.271865.27186
Root analytic conductor: 2.296052.29605
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ51(50,⋅)\chi_{51} (50, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 51, ( :2), 0.284−0.958i)(2,\ 51,\ (\ :2),\ 0.284 - 0.958i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.278552+0.207862i0.278552 + 0.207862i
L(12)L(\frac12) ≈\approx 0.278552+0.207862i0.278552 + 0.207862i
L(3)L(3) 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+(−1.27−8.90i)T 1 + (-1.27 - 8.90i)T
17 1+(262.+120.i)T 1 + (262. + 120. i)T
good2 1+7.45iT−16T2 1 + 7.45iT - 16T^{2}
5 1+18.0T+625T2 1 + 18.0T + 625T^{2}
7 1−41.9iT−2.40e3T2 1 - 41.9iT - 2.40e3T^{2}
11 1+94.6T+1.46e4T2 1 + 94.6T + 1.46e4T^{2}
13 1−128.T+2.85e4T2 1 - 128.T + 2.85e4T^{2}
19 1+489.T+1.30e5T2 1 + 489.T + 1.30e5T^{2}
23 1−827.T+2.79e5T2 1 - 827.T + 2.79e5T^{2}
29 1+955.T+7.07e5T2 1 + 955.T + 7.07e5T^{2}
31 1−1.63e3iT−9.23e5T2 1 - 1.63e3iT - 9.23e5T^{2}
37 1+657.iT−1.87e6T2 1 + 657. iT - 1.87e6T^{2}
41 1+846.T+2.82e6T2 1 + 846.T + 2.82e6T^{2}
43 1−2.59e3T+3.41e6T2 1 - 2.59e3T + 3.41e6T^{2}
47 1+1.07e3iT−4.87e6T2 1 + 1.07e3iT - 4.87e6T^{2}
53 1−1.46e3iT−7.89e6T2 1 - 1.46e3iT - 7.89e6T^{2}
59 1+3.91e3iT−1.21e7T2 1 + 3.91e3iT - 1.21e7T^{2}
61 1−1.06e3iT−1.38e7T2 1 - 1.06e3iT - 1.38e7T^{2}
67 1−531.T+2.01e7T2 1 - 531.T + 2.01e7T^{2}
71 1+3.80e3T+2.54e7T2 1 + 3.80e3T + 2.54e7T^{2}
73 1−2.28e3iT−2.83e7T2 1 - 2.28e3iT - 2.83e7T^{2}
79 1+391.iT−3.89e7T2 1 + 391. iT - 3.89e7T^{2}
83 1−1.02e4iT−4.74e7T2 1 - 1.02e4iT - 4.74e7T^{2}
89 1−2.94e3iT−6.27e7T2 1 - 2.94e3iT - 6.27e7T^{2}
97 1−1.19e4iT−8.85e7T2 1 - 1.19e4iT - 8.85e7T^{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

−14.92432560630545923402554602016, −13.45525418404069997461623416333, −12.39184634868798593880263228059, −11.15355793328455591739512043061, −10.72152750966606881748784931234, −9.186169939543145805759498359836, −8.523695046929503960857237312506, −5.15330970614554316404847802849, −3.88102743906127183741357547166, −2.56003560911344764538986072968, 0.20334938037297921677733269397, 4.17378667876476599411877365563, 5.98321647559786410143521429429, 7.12306092888502342353915379338, 7.917635515889894772996041177359, 8.885237429770323361510952320661, 11.00519649606674667462758869377, 13.07463193499122802834729723077, 13.36893807093085422350608890398, 14.79378290394242007732172863583

Graph of the ZZ-function along the critical line